The book of Romans is more than a letter from Paul to believers in Rome. It is a detailed work on the life of a Christian: both in what to believe and how to act on that belief.
Chapter 1 starts with some basic points: believers in Christ are "set apart" for the gospel (v. 1), that gospel was promised in scripture and through the prophets (v. 2), and that the Messiah was a proven descendant of King David (a big sticking point for the Jews). Jesus was also proven to be the Son of God by his resurrection from death.
These foundational truths are followed to include an acknowledgment that Christians are to call non-believers to the obedience that comes through faith. Note the uncomfortable there: obedience. Faith requires it.
Faith requires action. It requires us to do something. In Acts 6:7 you can read about many priests who obeyed when they became believers in Jesus. Having faith is one thing. Many will claim to have faith in Jesus. It's the obedience, however, that shows just how real that faith is.
For example, let's say you have a sore throat. You've had it for a few days, and it feels like you're downing a handful of nails every time you swallow. You go to the best ENT doctor around. He's got a great reputation and is extremely knowledgeable about all things throat-related. He looks at your throat, recognizes right away that you have strep, and gives you a prescription. He recommends you stay out of work the remainder of the week and rest. Good advice, right?
The problem is, you never get the prescription filled. You blow off his advisement that you get rest and you go back to work, pretending that no one will notice how rough you look as you waltz around with your 102-degree fever. You liked the doctor, you agreed with what he said, but you didn't trust him enough to follow his advice. So you keep going with a strep that makes you feel awful, despite knowing exactly what you should do to get better.
That's faith without obedience. That's us saying we trust God and we trust his word, but not to the point we're going to act on it. But our faith is evidenced by our willingness to follow who we profess to trust.
We don't like obedience for a litany of reasons, but mainly two that stick out. Obedience requires submission. It forces us to admit that we are under someone else's authority and to believe someone else may have a better idea as to what we should be doing than we do.
Secondly, obedience demands discipline. We set routines and make sacrifices of things we'd rather do so we can fulfill our obedience.
The obedience we are called to through faith is not one of legalism. Rather it is of love for our Father and his word. How is your faith? Would you consider it to be strong? If so, does your obedience show that?
The old hymn says "Trust and obey...for there's no other way...to be happy in Jesus...than to trust and obey." Obeying Jesus provides joy. Not the type so often seen in the world that is temporary and fleeting, but a joy that lasts. We can have joy in obeying God because it means we have disciplined ourselves to understanding his word and hearing him speak to us through it and our prayers, and in so doing we have proven the life-change faith in Christ produces.