I'm so jealous. There's so much to do there, but probably nothing I could suggest that isn't obvious: the Colosseum, the Pantheon (have to go inside, to see the oculus), the Catacombs, the Spanish Steps, the Trevi Fountain. The Vatican if only to see Michelangelo's La Pieta, which is one of the most beautiful things I've ever seen - photos don't do it justice. Would recommend the Piazza Navone/Navona (spellings differ) as someone to eat/have a snack (you'll probably recognise it from The Talented Mr Ripley and Roman Holiday). Speaking of Roman Holiday, there's the Mouth of Truth too. Maybe you should rent it before you go.

There are some beautiful parks (and theme parks!) too:
http://www.rome.world-guides.com/parks.html

Have been a few times - only experienced one (attempted) pickpocket, when a woman shoved a map in my Dad's face and tried to nick his wallet from his shirt pocket...