For me, no trip to Rome is complete without at least on dish of Saltimbocca ala Romana. It is among my favorite dishes on the planet.
The gelateria at Tre Scalini for tortuffo -- a blob of intensely dark chocolate gelato rolled in chopped dark chocolate and served with just a bit of whipped cream.
Alfredo's al Augusteo -- may seem like a cliche, but fettucine Alfredo at Alfredo's is on a different plane from anything called fettucine Alfredo in the U.S.
For history, Hostaria dell’Orso. It was Dante's favorite spot. Very elegant, reservations a must. The Hostaria dell’Orso is in a splendid, 14th-century building (and worth the visit, I think, just for the rare opportunity to eat in a building of such antiquity). On the ground floor, there is a lovely piano bar for a before-dinner drink. The restaurant is on the next floor up, divided between the level’s several large rooms.
Another fun choice is Sora Lella on the Isola Tiberina. This island in the River Tiber was built up by the Romans to resemble a ship, because that caused the water to pass around the island, rather than tear it apart. Sora Lella is a small trattoria, but it is very Roman, with daily specials such as maialino all'antica roma (suckling pig with prunes, pine nuts, and raisins) and Roman ricotta cake. It’s fun to visit the restroom, because the water of the Tiber River rushes past only a few feet below the window! Begun by a local woman, Sora Lella, the restaurant has become something of a monument to one of her sons, who became a widely known entertainer in Rome. It’s not as cheap as some trattoria’s, but it’s more interesting than most other trattoria’s.
Buon viaggio.