FOR I AM NOT ASHAMED: Ou gar epaischunomai (1SPMI):

For (1063) (gar) is a subordinating conjunction expressing cause or  explanation and thus introduces an explanation. He is explaining why he is eager to preach the gospel to the saints in Rome (notice the gospel is not just to "get one saved" but is actively involved in our ongoing day to day salvation from sin, Satan and self (see Three Tenses of Salvation)

Not (3756) (ou) indicates absolute negation and strongly denies the possibility that Paul might ever be ashamed of the glorious gospel. The verb ashamed is also in the present tense indicating this was Paul's continual attitude.

Ashamed (1870) (epaischunomai from epi = upon or used to intensify the meaning of the following word + aischunomai from aischos = disfigurement & then disgrace) (Click study of epaischunomai) (used 2x in Romans)  means to experience a painful feeling or sense of loss of status because of some particular event or activity. It describes one's consciousness of guilt or of exposure or the fear of embarrassment that one's expectations may prove false. Epaischunomai  is associated with being afraid, feeling shame which prevents one from doing something, a reluctance to say or do something because of fear of humiliation, experiencing a lack of courage to stand up for something or feeling  shame because of what has been done.

Writing to the Corinthian saints Paul explained that "we preach (kerusso = herald as a public crier) Christ crucified, to Jews a stumbling block (Greek = skandalon gives us our English "scandal" = circumstance or action that offends propriety or established moral conceptions), and to Gentiles foolishness (considered intellectually weak and irrational)" (1Cor 1:23) and thus we can see why the gospel might bring about situations in which one would might be tempted to feel a sense of shame.

Paul was indubitably unashamed and the gospel had indeed created  many "scandals"  for Paul - he had been imprisoned in Philippi (Acts 16:23, 24), chased out of Thessalonica (Acts 17:1-9), smuggled out of Berea (Acts 17:10-15), sneered at in Athens (Acts 17:32), regarded as a fool in Corinth (1Cor 1:18 23), and stoned in Galatia (Acts 14:19), but Paul remained eager to preach the gospel in Rome—the seat of contemporary political power and pagan religion. Neither ridicule, criticism, nor physical persecution could curb his boldness. (See 2Cor 4:5-18; 11:23-28; 12:9). But its inherent glory, as God's life-giving message to a dying world, so filled his soul, that, like his blessed Master, he "despised the shame." (see note Hebrews 12:2)

Paul knew that Rome was a volatile place and that Christians there had already experienced persecution. He knew that the capital city of the empire was steeped in immorality and paganism, including emperor worship. He knew that most Romans would despise him and that many probably would do him harm. Yet he was boldly eager to go there, for his Lord’s sake and for the sake of the Lord’s people.

He was not ashamed even though he had been imprisoned in Philippi, chased out of Thessalonica, smuggled out of Damascus and Berea, laughed at in Athens, considered a fool in Corinth, and declared a blasphemer and lawbreaker in Jerusalem. He was stoned and left for dead at Lystra. Some pagans of Paul’s day branded Christianity as atheism because it believed in only one God and as being cannibalistic because of a misunderstanding of the Lord’s Supper. Although that GOSPEL was then, and still is today, a stumbling block to Jews and foolishness to Gentiles, it is the only way God has provided for the salvation of men, and Paul was both overjoyed and emboldened by the privilege of proclaiming its truth and power wherever he went.

The fellowship of the unashamed - When we have opportunity to speak for Christ, we often do not. We know the gospel is unattractive, intimidating, and repulsive to the natural, unsaved person and to the ungodly spiritual system that now dominates the world. The gospel exposes man’s sin, wickedness, depravity, and lostness, and it declares pride to be despicable and works righteousness to be worthless in God’s sight. To the sinful heart of unbelievers, the gospel does not appear to be good news but bad, and when they first hear it they often react with disdain against the one presenting it or throw out arguments and theories against it.  Fear of men and of not being able to handle their arguments are some of the greatest impediment to being a bold witness for the gospel of Jesus Christ.

It is said that if a circle of white chalk is traced on the floor around a goose that it will not leave the circle for fear of crossing the white mark. In a similar way, the "chalk marks" of criticism, ridicule, tradition, and rejection prevent many believers from leaving the security of Christian fellowship to witness to the unsaved.

Jameison writes that Paul's

language implies that it required some courage to bring to "the mistress of the world" what "to the Jews was a stumbling-block and to the Greeks foolishness" (1Co 1:23). But its inherent glory, as God's life-giving message to a dying world, so filled his soul, that, like his blessed Master, he "despised the shame."

Warren Wiersbe gives a personal illustration of why Paul was not ashamed:

During my years in high school, I was chosen to be an office monitor. The other hall monitors sat at various stations around the building, but I was privileged to sit right outside the door of the main high school office. I was entrusted with important messages that I had to deliver to different teachers and staff members, and on occasion even to other schools. Believe me, it was fun to walk into a classroom and even interrupt a lesson! No teacher ever scolded me, because all of them knew I carried messages from the principal. I never had to be afraid or ashamed, because I knew where my messages came from.

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Unashamed - On one occasion Frederick the Great invited some notable people to his royal table, including his top-ranking generals. One of them by the name of Hans von Zieten declined the invitation because he wanted to partake of communion at his church. Some time later at another banquet Frederick and his guests mocked the general for his religious scruples and made jokes about the Lord’s supper. In great peril of his life, the officer stood to his feet and said respectfully to the monarch, “My lord, there is a greater King than you, a King to whom I have sworn allegiance even unto death. I am a Christian man, and I cannot sit quietly as the Lord’s name is dishonored, His character belittled, and His cause subjected to ridicule. With your permission I shall withdraw.” The other generals trembled in silence, knowing that von Zieten might be killed. But to their surprise, Frederick grasped the hand of this courageous man, asked his forgiveness, and requested that he remain. He promised that he would never again allow such a travesty to be made of sacred things. (Our Daily Bread, Copyright RBC Ministries, Grand Rapids, MI. Reprinted by permission. All rights reserved)

OF THE GOSPEL: to euaggelion:

Gospel (2098) (euaggelion from eú = good + aggéllo = proclaim, tell) (Click in depth study of euaggelion) was originally a reward for good news and later became the good news itself. The word euaggelion was in just as common use in the first century as our words good news today. “Have you any good news for me today?” would have been a common question. In this secular use euaggelion described good news of any kind and prior to the writing of the New Testament, had no definite religious connotation in the ancient world until it was taken over by the "Cult of Caesar" which was the state religion and in which the emperor was worshipped as a god

Paul's "definition" of the gospel is summarized in the following passage (as you present the gospel be sure to include these foundational truths)

1 Now I make known to you, brethren, the gospel which I preached to you, which also you received, in which also you stand,
2 by which also you are saved, if you hold fast the word which I preached to you, unless you believed in vain.
3 For I delivered to you as of first importance what I also received, that Christ died for our sins according to the Scriptures,
4 and that He was buried, and that He was raised on the third day according to the Scriptures,
5 and that He appeared to Cephas, then to the twelve.
6 After that He appeared to more than five hundred brethren at one time, most of whom remain until now, but some have fallen asleep;
7 then He appeared to James, then to all the apostles;
8 and last of all, as it were to one untimely born, He appeared to me also.
9 For I am the least of the apostles, who am not fit to be called an apostle, because I persecuted the church of God.
10 But by the grace of God I am what I am, and His grace toward me did not prove vain; but I labored even more than all of them, yet not I, but the grace of God with me.
11 Whether then it was I or they, so we preach (the gospel which has inherent power and do so by the empowering grace of God that works within us) and so you believed. (1Corinthians 15:1-11 )

Paul had explained earlier in this same letter to the church at Corinth that proclamation of the gospel was his primary mandate...

17 for Christ did not send (apostello - our word "apostle") me to baptize, but to preach the gospel, (how?) not in cleverness of speech, (why not use "clever speech"?) that the cross of Christ (in context synonymous with the gospel) should not be made void.
18 For the word of the cross (again synonymous with the gospel) is to those who are perishing (not being annihilated but suffering eternal loss and ruin, no longer ever able to be useful for the purpose for which they were created!) foolishness (this is why Paul said he was not ashamed of the gospel for the wise would try to shame those who proclaim such a foolish message), but to us who are being saved it is the (inherent ability) power of God .
19 For it is written, "I WILL DESTROY THE WISDOM OF THE WISE, AND THE CLEVERNESS OF THE CLEVER I WILL SET ASIDE."
20 Where is the wise man? Where is the scribe? Where is the debater of this age? Has not God made foolish the wisdom of the world?
21 For since in the wisdom of God the world through its wisdom did not come to know God, God was well-pleased through the foolishness (Paul is being sarcastic and explains that God's "foolishness" is wiser than man) of the message preached to save those who believe.
22 For indeed Jews ask for signs, and Greeks search for wisdom;
23 but we preach Christ crucified (the heart of the gospel message), to Jews a stumbling block, and to Gentiles foolishness,
24 but to those who are the called, both Jews and Greeks, Christ (and the good news about Him which is ) the power of God and the wisdom of God.
25 Because the foolishness of God is wiser than men, and the weakness of God is stronger than men.  (1Corinthians 1:17-25)

Paul again explained his purpose writing

2:1 And when I came to you, brethren, I did not come with superiority of speech or of wisdom, proclaiming to you the testimony (marturion) of God (the gospel, cf "the testimony of our Lord" see discussion 2 Timothy 1:8).
2 For I determined (resolved) to know nothing among you except Jesus Christ, and Him crucified.
3 And I was with you in weakness (state of incapacity to do something = we like Paul should be so "weak" that our only power is from God) and in fear and in much trembling (quaking or quivering with fear but still unashamed).
4 And my message (logos) and my preaching (proclamation) were not in persuasive words of wisdom (as men would use in attempting to win an argument or debate), but in demonstration of the Spirit and of power (perfect parallel of Romans 1:16 emphasizing the intrinsic supernatural, divine power of the gospel),
5 that your faith should not rest on the wisdom (sophia) of men (earthly, natural, unspiritual, fleshly, even motivated by the Devil), but on the power of God (the gospel of God).
6 Yet we do speak wisdom among those who are mature (see teleios); a wisdom, however, not of this age, nor of the rulers of this age, who are passing away (not so much simply dying but katargeo = basic idea is they are useless and ineffective);
7 but we speak God's wisdom in a mystery, the hidden wisdom, which God predestined before the ages to our glory (He made it for our benefit before the world began);  (1Corinthians 2:1-7)

Paul considered himself first as a servant of Christ with his call to proclaim the gospel as a stewardship...

1 Let a man regard us (indicating Paul, Apollos, Cephas, and, by extension, all other “fellow–workers”) in this manner, as servants (huperetes = “under rowers” = lowest galley slaves rowing on bottom tier of ship = most menial, unenvied, despised of slaves = thus subordinates of any sort, to those under the authority of another) of Christ (servants cannot serve men rightly unless they serve their Lord rightly and they cannot serve Him rightly unless they see themselves rightly = as His under-slaves, His menial servants - he serves Christ first and then and only then can he best serve people), and stewards (responsible for the Master's "property" and one day to give an account) of the mysteries (musterion = in NT that which was previously hidden but now made known only by divine revelation) of God.
2 In this case, moreover, it is required of stewards (oikonomos = “house manager” = person placed in complete control of household supervising property, fields, finances, etc on behalf of his master)  that one be found trustworthy (faithful). (1Corinthians 1:17-25)

 Finally in (1Corinthians 9:16-18) Paul explains why he must preach the gospel...

16  For if I preach the gospel, I have nothing to boast of, for I am under compulsion (if someone is compelled they are driven irresistibly); for woe is me if I do not preach the gospel.
17 For if I do this voluntarily, I have a reward; but if against my will, I have a stewardship (responsible management of the gospel entrusted to Paul’s care) entrusted to me.
18 What then is my reward? That, when I preach the gospel, I may offer the gospel without charge, so as not to make full use of my right in the gospel.

The so-called "health and wealth gospel" that has swept through much of the church today is not offensive to the world because it offers what the world wants. But that false gospel ("a different [heteros] gospel which is really not another" Galatians1:6-7) does not offer the gospel of Jesus Christ. What does the gospel do? It saves sinners. What else will save sinners? Not science, not education, not religion, not moral reformation, not fame and fortune. The gospel, and only the gospel, saves sinners.

William Tyndale (1494–1536) who was martyred because he was not "ashamed of the gospel" said 

'Euaggelion (which we call gospel) is a Greek word, and signifies good, merry, glad, and joyful tydings, that makes a mans heart glad, and makes him sing, dance, and leap for joy.'

John MacArthur writes that...

These two verses express the theme of the book of Romans, and they contain the most life-transforming truth God has put into men’s hands. To understand and positively respond to this truth is to have one’s time and eternity completely altered. These words summarize the gospel of Jesus Christ, which Paul then proceeds to unfold and explain throughout the remainder of the epistle. Some years ago after speaking at a youth rally, the wife of the rally director approached MacArthur and said, “Your message offended me, because you preached as if all of these young people were sinners.” To which he replied “I’m glad it came across that way, because that is exactly the message I wanted to communicate.”

Paul’s supreme passion was the GOSPEL (set apart for it in Ro1:1) & thereby to see men saved. He cared little for personal comfort, popularity, or reputation. He offered no compromise of the gospel, because he knew in the GOSPEL was the only power available that can change lives for eternity.

Geoffrey Wilson wrote,

The unpopularity of a crucified Christ has prompted many to present a message which is more palatable to the unbeliever, but the removal of the offense of the cross (1Co1:18) always renders the message ineffective. An inoffensive gospel is also an inoperative gospel."

Simply scanning the main subjects in the four main divisions of Romans (see table) leaves no doubt that the presentation of the gospel begins with the "bad news" to awaken in the hearer the fact that unbelievers need God's righteousness.

FOR IT IS THE POWER OF GOD: dunamis gar theou estin (3SPAI): (Illustration) (Power of the gospel) (Illustration)

Click the 12 occurrences of the phrase "power of God" in NT.

Spurgeon comments on this power noting that...

It is a wonderful heart-searching text, is this: let us put ourselves under its power. Whatever you obey, that is your master: and if you obey the suggestions of sin, you are the slave of sin: and it is only as you are obedient to God that you are truly the servants of God. So that, after all, our outward, walk and conversation are the best test of our true condition. Without holiness no man shall see the Lord, nor can he have any reason to believe that he belongs to God.

The power of God - As Morris says...

The gospel is not advice to people, suggesting that they lift themselves. It is power. It lifts them up. Paul does not say that the gospel brings power, but that it is (present tense = continually) power, and God’s (omnipotent) power at that.  (Morris, Leon. The Epistle to the Romans. Page 68. Grand Rapids, Mich.; Leicester, England: W. B. Eerdmans; Inter-Varsity Press. 1988 or Logos) (Bolding, italics and words in parentheses added)

Power was an attribute that certainly characterized the Roman Empire in Paul's day, but their power was only human power and the "powerful" Romans like all men of all ages were powerless to make themselves righteous before a holy God! Seneca in fact called Rome a "cesspool of iniquity" and Juvenal was not much kinder referring to Rome as a "filthy sewer into which the dregs of the empire flood." And so "powerful" Rome, like all men who are born into Adam (see note Romans 5:12) was in desperate need of the Gospel and the righteousness of God therein revealed!

Power (1411) (dunamis - words derived from the stem duna- or dyna- all have the basic meaning of “being able,” of “capacity” in virtue of an ability) is a key word in the NT being found some 121 times in the NASB most often in the Gospels.

Dunamis is the root from which we derive the English word dynamic, (synonyms = energetic, functioning, live, operative, working) which describes that which is marked by usually continuous and productive activity or change. That which is dynamic is characterized by energy or forces that produce motion, as opposed to that which is static. Another English word dynamite, is derived from dunamis and since dunamis is used by Paul to describe the "power of God", some have suggested that the gospel is "God’s dynamite". This is misapplication of this English derivative in an attempt to try to picture the life saving power of the gospel. Dunamis does not refer to explosive power, as if the gospel will blow men to bits but as discussed above, it refers to intrinsic power. The gospel is dynamic, God’s dynamic, and so is powerful and able to effect radical regeneration of spiritually dead men and women. The gospel makes dunamis power available to all believers.

Paul explained that...

the kingdom of God does not consist in words but in power (1Cor 4:20)

Paul's confidence in the gospel was based on its supremacy. He knew it was far superior to any religion or philosophy every known on earth. The world of Paul's day was dominated by three lines of thought -- Greek, Roman and Hebrew, but Greek logic, Roman law and Hebrew light all paled before the gospel.

People have an innate desire to be changed and many want to be changed inwardly, in a way that will make them feel less guilty and more content and so go to a host of programs, philosophies, and religions that promise to meet those desires. But although these "methods" may ostensibly succeed in making people feel better about themselves, the ideas promoted have no power to remove the SIN that brings the feelings of GUILT & DISCONTENT. Nor can those ideas make men RIGHT WITH GOD (righteous). In fact, the tragedy is that the more "successful" such approaches are, the more they drive people away from God and insulate them from His salvation. (cp futile speculations followed by a heart that becomes darkened to life giving spiritual truth).

MacArthur:

Scripture certainly testifies to God’s glorious power (Ex 15:6), His irresistible power (Dt 32:39), His unsearchable power (Job 5:9), His mighty power (Job 9:4), His great power (Ps 79:11- Spurgeon's note), His incomparable power (Ps 89:8 - Spurgeon's comment), His strong power (Ps 89:13 - Spurgeon's note), His everlasting power (Isa 26:4), His effectual power (Isa 43:13), and His sovereign power (see note Romans 9:21). Jeremiah declared of God, “It is He who made the earth by His power, who established the world by His wisdom” (