Less frequented places:
Villa Borghese, with full of fountains, small buildings, and various attractions. And a copy of the "Globe" (as in Shakespeare + Theatre) in real wood. Never been in there. I suppose one watches the show while standing up, like in old Bard times.
Pincian hill - Pincian hill garden. Communicating with Villa Borghese through a bridge over the Aurelian walls. Same as above.
Janiculum hill - it is fundamental to take a history book and study about the Second Roman Republic, 1849. This place exudes those events and the fights against the bloody French accomplices of the bloody pope (said without resentment toward the French of today, of course).
The Appia Antica, old Appian way. You need to have good legs and plenty of will to put them at use. There's also a specific bus making several stops at various places. Absolutely visit some catacomb (San Callisto, or Santa Domitilla, being the obvious ones) see the tomb of Caecilia Metella (or just affectionately called Cecilia Metella) and obviously Porta San Sebastiano, Arco di Druso, the nearby Porta Latina and the nearby church San Giovanni a Porta Latina and "martyrium" San Giovanni in Oleo. (You can see Porta di San Sebastiano, Porta Latina, Arco di Druso and nearby churches on foot before taking the Appian bus).
Piazza Navona, Piazza di Spagna, Piazza del Popolo, Piazza San Pietro, the four major basilicas, Fontana di Trevi, Piazza di Spagna.
Then you have one museum ticket with which you can visit Colosseum, Roman Forum and Palatine hill in different days.
Do have a walk on the Caelian hill during the day, and another walk on the Caelian hill during the night, with your tripod.
If you still have time, just go anywhere in between those places and you won't be disappointed :-)
PS Buy a good guide, and study it well before going to Rome. You won't have time to read about anything. It will just slow you too much. Have a general reading of all the important stuff before. Take virtual walks with Google Maps. Learn the relation between the various hills and quarters. Walk, walk, walk.
Absolutely don't eat at restaurants which don't have the price list outside. Have a good guide to advice regarding restaurants, or ask the locals for reliable places.
Be extremely wary of gipsy children, typically waiting for you in train stations or in the street. They can surround you and deprive of your money much, much beyond your assumptions*.
* In case you think this remark is politically incorrect, then by all means do please ignore it.