You should take a look at the Trevi fountain - it's gorgeous. If you throw a coin into the fountain, it means that you'll return to Rome someday. Around the corner from the Trevi fountain, there's a shop that sells over 100 flavours of Italian ice-cream, which is a great place for a stop on a hectic day of sight-seeing! Hope you enjoy your trip!
I come from Rome and, apart from the very good suggestions other people have made, one of my favourite places is the Aventine Hill; there are some old churches and a good view of town, I find it beautiful and relaxing for a walk away from the mass tourist joints.