It is very helpful to slow down and take a good look at Paul's prayers. When we do, we learn some really crucial things. For instance, in Romans 15:13, Paul writes:
May the God of hope fill you with all joy and peace in believing, so that by the power of the Holy Spirit you may abound in hope.
First we see here that God is the God "of hope." God is the target of our hope - the One our hope rests on. And He is the source of our hope. So then, Paul tells us _how_ God gives us hope: by giving us joy and peace. How do we gain joy and peace? By believing. Believing what? If Paul were here right now, I bet the money in my pocket he would say, "By believing everything I've told you so far in this letter."
To put that another way: by believing the gospel.
Now, remember that Paul is writing to Christians. This is not evangelism. Paul is saying, "Christian, your food for joy and peace is the gospel, as I have laid it out here in this letter. And you eat it by believing it, today." When you read the gospel in Romans, you discover that it is like a diamond, with many facets. Or, to continue the food metaphor, it is sometimes like meat and potatoes, while other times it's like crab - you have to work a little harder to get it open and enjoy the buttery goodness.
Either way, there is some aspect of the gospel that you and I need to be believing today. You might doubt your identity; you need to believe that you have died to sin and are alive to God in Christ (6:12). Trying really, really hard in your faith and getting nowhere? You need to chew on the frustration of end of chapter 7, and ask yourself where doing it by your own power is getting you. Struggling with sin? You may start at 8:1 and simply rest there a bit, and then get back up and put a sword through your lusts (8:13). Suffering? You need to believe 8:17 to the end of the chapter. Got questions about the Jews? Chapters 9-11 are your bag. Relationship struggles? You need to be believing 13:8 through chapter 14. Got questions about government? Romans 15, man. All of it flows from the gospel.
One more thing. How often have you enjoyed a fine steak or seafood meal in a fine restaurant, totally alone? Hopefully rarely, if ever. A fine meal is meant to be enjoyed together. So is the gospel, among Christians. The two "you"'s in this passage are plural. (We cannot tell this in English, but in the original language it was very clear.) That's because we (the church) are a "people", formed by a covenant: the new covenant, the gospel.
Thus we are meant to enjoy the gospel together. It’s at this point that we must remember that we have been reading a prayer. Paul prayed this for the Romans. We should pray this way for ourselves and for each other, that God would grant us authentic joy and peace in believing.
The Spirit then takes this joy and peace and produces in us a confidence that God will not leave unfinished what He started at that gory cross and that empty tomb. Our spines begin to joyfully stiffen with the hope that He will soon enough return and raise us to be like His Son. Joy and peace in believing causes us - by the Spirit - to abound, overflow in hope.