I lived and studied in Rome, as always it depends how long you're there for. I always used to spend time at Caracalla's Baths, as it was relevant to my thesis.
Aside from the usual, the Ara Pacis is in a nice new museum and usually quiet, with Augustus' Mausoleum next door (wish they'd do something with that).
The temples at Largo Argentina are worth a look.
In the Vatican itself, the Etruscan Museum is wonderful, but closed a lot of the time.
Wandering through the market section of Trajan's Forum is always pretty special.