I second the Galleria Borghese - booking essential.
For a luxurious, historical cafe go to Antico Caffe Greco in the Via Condotti.
Santa Maria del Popolo for the Caravaggios.
Get the lift to the top of the Vittorio Emmanuele monument for some good views.
If you're going to the Forum, make sure you visit the Palatine Hill too - it's much quieter.
Definitely recommend the San't Ignazio church - the baroque ceiling is stupendous.
Ara Pacis is impressive.
Piazza dei Fiori is a nice market, plus lots of restaurants.
Jewish quarter for authentic Jewish food - not far from the Pantheon.
You can walk everywhere, but group the things you want to see on separate days - so Castel Sant'Angelo with the Vatican, or Colosseum with the Forum. Otherwise you'll get in a real mess.
Remember that not only was Rome not built in a day, it can't be seen in one either. Nor in a week, nor a month. I have been 8 times, and there are still places/things I haven't seen. So be very selective, don't do too much, and take time to just watch the world go by.