Eat at Ristochicco. It's at Borgo Pio 186, between the Vatican and Castel Sant'Angelo. The place is outstanding, and a terrific value. It's quite easy to overspend while in Rome, particularly on the food, but this place will fill your belly with some of the best food I've ever tasted (the gnocchi was phenomenal) and fill your heart with some very engaging and kind service. This place was fantastic, and without question the best meals we had in Rome (we ate there twice, we enjoyed it so).
I would also suggest trying La Famiglia, near the Terminii on Via Gaeta. It's a good restaurant, though it seemed more of a tourist place to me. Still, their pasta was very good, in my opinion. I had a stuffed tortelini in a great cheese sauce. All things considered, quite good.
And while we were there, I found that I wasn't as hungry as I tend to get at home, though I was moving a whole lot more - we walked like it was going out of style. Plan on lots of hills when going to Rome, and streets that are not terribly smooth - they pave with basalt cobblestones called "sampietrini" by the locals - literally "Little stones of St. Peter." You'll be tired at the end of the day, but you'll have seen quite a bit.