From the first chapter of the epistle to the Romans beginning in verse one

The beginning is a very good place to start. So in my quest through the epistles, I begin with the apostle Paul's letter to the Romans, which begins with an introduction to the author and a statement of the original audience:

1 Paul, a servant of Christ Jesus, called to be an apostle, set apart for the gospel of God, ² which he promised beforehand through his prophets in the holy Scriptures, ³ concerning his Son, who was descended from David according to the flesh ⁴ and was declared to be the Son of God in power according to the Spirit of holiness by his resurrection from the dead, Jesus Christ our Lord, ⁵ through whom we have received grace and apostleship to bring about the obedience of faith for the sake of his name among all the nations, ⁶ including you who are called to belong to Jesus Christ,
⁷ To all those in Rome who are loved by God and called to be saints:

The apostle Paul puts his name to the letter, which helps us as readers. Those who have been reading the scriptures straight through or have been exposed to the Scriptures for a while will remember that Paul became a Christian in a rather spectacular show of God's power and love. I will not retell the story, but will cite it: Acts IX.i-xix.
After giving his name, Paul kind of gives us an overview of his purpose after his conversion. He says first and foremost that he is a servant of Christ. Paul has dedicated his life to serving Christ and he puts that right out there for everyone to see. Then he says that he is called to be an apostle. This is a specific place in the early church that was responsible for teaching and discerning doctrine in the early church. He tells us a little bit about being an apostle in the next phrase as he says also that he is "set apart for the Gospel of GOD".
Later on, he will get around to explaining exactly what that Gospel is, but right here he explains that that Gospel was promised before the coming of Christ. Throughout the Old Testament and the period of the Law of Moses, God announced that Christ was coming. The Jewish people knew that a Saviour was coming, which makes the next phrase all that more important to understand. Paul comments that this Gospel that was promised is all about his Son. We know that this must be Christ who identifies as the son of God, even though he came from the line of David. There is an important distinction we must know about Christ made here: Christ in flesh and Christ in power.
Christ came as a descendent, a great-great-great-great grand son, of David. That was his physical flesh, but his power was from the Spirit that we understand to be God because of his resurrection. We know that God is the only one who can raise from the dead, so this is a clear claim to the power of God for Christ. So far, the apostle Paul has been talking about the nature of Christ. How he was promised before, he came from David's family, and had the power of God. Christ was very unique in this. His resurrection sets him apart from the rest of humanity.
Paul goes on to comment that it is Christ that he received grace and the power to be an apostle, which we discussed earlier. This power of apostleship, Paul Says, allows him to teach others to be obedient to Christ. The reason that we as believers are to be obedient is to promote the name of Christ in the "nations". The nations as used here would include all of humanity. He makes a special comment about those who are called to Christ. That would be the church that the apostle Paul is writing to.
Paul says that it is Christ through whom we get grace and he gets that authority to teach obedience to Christ as an apostle.
The audience is the believers in Rome, those who have already trusted in the grace of the Gospel. It should be noted that Paul does not identify a single, unified church, but all those who believe. Chapter sixteen, at the end of the book, identifies the church of Prisca and Aquila in their house among his personal greetings. It is not for certain, but these indicate that Rome may have had many small house churches at the time of this letter.
All of this has been building up to this: 

Grace to you and peace from God our Father and the Lord Jesus Christ.

In verse eight, Paul makes an interesting comment:

⁸ First, I thank my God through Jesus Christ for all of you, because your faith is proclaimed in all the world.

He says that he thanks God for the church in Rome. Paul is grateful for them because their faith is famous all over the world. Of course, we are speaking only the Roman world, but that is still an accomplishment. In the day, Rome was the centre of politics and culture and for the early church to have a clearly established presence there was a big deal.

⁹ For God is my witness, whom I serve with my spirit in the gospel of his Son, that without ceasing I mention you ¹⁰ always in my prayers, asking that somehow by God's will I may now at last succeed in coming to you.

The spiritual sense with which the apostle Paul lives his life is made even more clear in these verses. He calls of God to be his witness, meaning that he is really serious about what he is about to say. Then he qualifies, or gives a little description of, who God is. God is the one that Paul serves, which he brings out earlier when he calls himself a servant of Christ. Then he comments that he prays without ceasing, that is stopping, for something. That is he will be able to come to them. The apostle Paul made it a regular habit to go around to the multiple churches and visit them. Often, when he couldn't, he would write them a letter like this. Paul still wants to visit, and he goes on to explain why:

¹¹ For I long to see you, that I may impart to you some spiritual gift to strengthen you— ¹² that is, that we may be mutually encouraged by each other's faith, both yours and mine.

At first, he says something that seems a little weird, to give a spiritual gift to the believers in Rome? We know that the Holy Spirit is the one who gives out spiritual gifts, generally not the apostles. He clarifies in verse 12 when he says that he wants to allow them to encourage each other. Paul wants to go to the church at Rome so that he can see their faith and get hope from that, and they can see his faith and get hope from that. He continues explaining how he wants to come and visit the church at Rome:

¹³ I want you to know, brothers, that I have often intended to come to you (but thus far have been prevented), in order that I may reap some harvest among you as well as among the rest of the Gentiles.

"Reap some harvest"? This phrase seems really odd at first, but we remember back to Luke X.ii where Our Lord himself makes a comment about how the harvest is plentiful. This has been understood to mean that there are many who are unsaved who are to get saved. Paul knows this and makes a reference to that as he says that he wants to come to Rome and share the Gospel there. However, in verse 14, Paul begins to explain why he hasn't been able to come:

¹⁴ I am under obligation both to Greeks and to barbarians, both to the wise and to the foolish.

Most translations say "obligation", and the Authorised Translation says "debtor". I do not believe that this should be taken to mean that Paul owes them money, instead, we should look at how Paul makes the two divisions. Between "Greeks" and "barbarians", that covers the whole of the world, as does "wise" (sometimes understood to be "educated") and "foolish" (likewise "uneducated"). This might be taken to mean that Paul is obligated to preach the Gospel to all people. That as much as he wants to preach the Gospel in Rome, there are other people who need his preaching too.

¹⁵ So I am eager to preach the gospel to you also who are in Rome. ¹⁶ For I am not ashamed of the gospel, for it is the power of God for salvation to everyone who believes, to the Jew first and also to the Greek.

I will admit upfront that verse 16 is often taken right from it's proper context and shoved into whatever mold a speaker wants to fit it in. In this sense, Paul is trying to explain why he want to preach in Rome so badly. It is because he is proud of the Gospel. He wants everyone to know because of how powerful it is. The Gospel is the saving power of Christ for all that believe and Paul makes another distinction that ends up covering the whole world: Jews and Greeks. The fact that it was to the Jews first is a reference to the fact that originally, the Gospel was for the Jews, but when they rejected it, the Gospel went to the Greeks, the gentiles. And then, the apostle Paul makes a comment about the nature of the Gospel:

¹⁷ For in it the righteousness of God is revealed from faith for faith, as it is written, “The righteous shall live by faith.”

He says that the Gospel is how God shows his righteousness, that is freedom from sin. The righteousness is "from faith to faith", or it starts with faith and is meant for faith. Paul then quotes Habakkuk II.iv in saying “The righteous shall live by faith.” This is important because this simple phrase describes what God finds pleasing "faith".

Even though this section may seem to be just the intro and the precursor to the book of Romans, we can learn a lot. Paul gives a quick overview of Christ which gives us a succinct statement of who it is we believe in. Paul then expresses his desire to come and preach, even though he has tried and can't come yet. In this, we understand the character of Paul a little better. He makes it clear that he just wants to keep preaching and gives us his reason. That the Gospel is the very power of God. This is the same Gospel that we have, that we profess. This is the same Gospel that other sections of Romans is famous for giving a good overview of.