I would also agree with La Tagliata . We first went there years ago when recommended by our driver Gianluca. He took us back in to the kitchen after the meal and we met "Mama." Great food, good value, incredible views, and fun experiences each time. If you've got deep pockets, the dining room at La Sirenuse - La Sponda - is wonderful. Quite romantic... and yes, quite expensive!
In Rome, I'd walk away from the Trevi a few blocks. Up the hill, I like Colline Emiliane. It's tiny, almost a hole in the wall on Via Avignonesi - an alleyway! Great food...
Heading in the other direction, toward the Pantheon... I like Da Gino. Small, and always crowded, I often go here for lunch, but you'll need a reservation, as they are close to parliament and have limited seating. I think they have some of the best pasta in town. It's even recommended by Condé Nast Traveler!
And for a little fancier place, I always eat at Armando Al Pantheon. Always consistent, always good - but closed on weekends!
None of these have outdoor seating or tourist menus - just good food!