Here is the heart of the argument: "we are justified by faith and now have wholeness (peace) with God through our Lord Jesus Christ." (Romans 5:1) Paul notes immediately that this is but the beginning of our journey of faith. It is often said in evangelical circles that the goal of the Christian faith (the goal of the Church) is to get everyone to heaven -- that is to have them make a commitment to Jesus Christ, receive the forgiveness for their sins and plan to live forever in heaven. This teaching is dangerous and stops short of the full gospel message. That commitment to Christ is but the beginning step down the road to discipleship. Jesus never says "Go and make church members" or "Go and make followers" he says "go and make disciples" (committed/disciplined followers of Jesus Christ). Romans 5:3 and following shows part of this journey: we rejoice in Hope, but we rejoice in our difficulties because these difficulties are what form and shape our character and it is this redeemed character that allows us to not only endure but to grow and thrive throughout this life.
The end of the chapter (12 to the end) is an explanation of how the fall of Adam is now corrected by the rising of Jesus Christ. In Adam the human race fell into a condition where our original condition (in the image and likeness of God) has been shattered and broken. Since we all come from Adam we all share that shattered image. Jesus, a kind of "second" Adam, restores the image and likeness of God within us through our faith and trust in him.