First off, RS’s time-saving, crowd-beating tips will be critical for you to get the most out of a short time.
1) Vatican Museum and Sistine Chapel: Get early morning tickets online, skip all the lines and enjoy it before it gets SUPER crowded. If you can, sneak into St. Peter’s from the Chapel. If you can’t, do St. Peter’s on the next trip.
2) See the Forum/Colosseum/Capitoline Hill: I would advise skipping the tour of the Colosseum. It’s best view is from the outside. Touring the inside takes time to negotiate, and perhaps controversially, I felt it was not worth it. If you’re feeling up to it, the Capitoline Museum is very good (and air-conditioned.)
3) Rick Steve’s “Heart of Rome” walk is the best. In between the crowds of tourists, you’ll find it's still a vibrant neighborhood of locals (maybe less so nearer the Trevi Fountain). Make extra time to go inside the magnificent churches along the way. This was perhaps my favorite day in Rome!
4) Borghese Gallery: It is second to none, and I can’t recommend it enough. You have to buy tickets well ahead of time, but this makes it something you can really plan your day around. The park itself is underwhelming. Taxi there and back.
In the evening, I’d suggest a food walking tour! I did one in both Trastevere and Testaccio. They were terrific introductions to Roman cuisine, the neighborhoods, and their unique histories. My only regret was not having another night or two to follow-up these places on my own (and with another stomach.) I’m looking forward to exploring both of these neighborhoods more next time.