Like everyone has said, good food is everywhere in Rome. I liked these two places a ton. Trattoria da Alfredo e Ada can get busy and the place is small, but the menu changes daily and is pretty inexpensive. It's super low key and feels very local, though more and more tourists are going. Incredible asparagus and artichoke lasagna when in season. Everything is good though.
Gelateria dell'Angeletto is superb gelato. Well priced (I mean, all gelato there is cheap) and incredibly fresh tasting. I know gelaterias are a dime a dozen, but IMO, this is the best I've had in Rome.