I grew up in Rome, so I'm often the first person people text when they plan a trip. I spent most of my time in Italy in front of a computer screen, so I don't have a super extensive list! There might be a lesson in there about appreciating what you have :)
Clearly the most important part of your trip. You can find really good pasta in the US, but really good pizza is actually hard. All of these places aren't in the touristy neighborhood, but worth the trip if you're close by for other reasons.
Fyi, Pizza in Rome is thin and crunchy, while Neapolitan pizza has a thicker crust and a softer dough. There's also "pinsa" which is another traditional Roman type of pizza.
Pizzeria L'Economica in San Lorenzo. My grandpa grew up in that neighborhood, which was one of the few bombed in 1943 during World War II. In 1946, L'Economica first opened and we've celebrated many family birthday parties there since then. There are 4 historic pizza places in the neighborhood, with the other three being Il Podista, Formula 1, and the now-closed Il Maratoneta (which used to be make amazing XL pizzas). The neighborhood can be kinda sketchy and rowdy during the weekend, so either enjoy the adventure or just leave after dinner!
Il Giardino Segreto by Tuscolana. They only make gigantic pizzas. The restaurant is hidden behind a wall of plants and the aqueducts ruins on the other side. They have a ton of outdoor seating and it's really quiet outside of some vibrations from the subway right below it. We did my aunt's reharsal dinner here!
Pizzeria Ostiense in Ostiense. Sit down, write your order on a piece of paper, eat quickly and leave :) It's a very high turnover restaurant so don't expect any fancy service but the food is great. This is a extremely busy place after every AS Roma game with people coming to get food after the stadium.
Dar Poeta in Trastevere. I don't go very often as it's right in Trastevere, but it's a classic.
Buca di Ripetta by Piazza del Popolo; I had dinner with my wife here before I proposed to her, so it's a special place for us. They have pretty good food but most importantly fair prices for the neighborhood.
Tonnarello in Trastevere; they have quick service, nice outdoor seating and good handmade pasta with traditional Roman dishes (Carbonara / Amatriciana / Gricia)
La Sora Lella in Trastevere. "Sora Lella" was the nickname of Elena Fabrizi, a very famous Roman actress who was the grandmother of the now-owner of the restaurant. She's one of the "quintessential Roman" people, so this place has a special place in everyone's heart.
Contrario at the Colosseum; it's not easy to find good places right in the center but this is definitely one of them!
Le Terrazze al Colosseo at the Colosseum; very nice place with a terrace overlooking the Colosseum (reserve in advance to sit on that side). We went to dinner here with my wife to celebrate when I joined Decibel!
Colline Emiliane near Barberini; awesome Michelin-star restaurant with very affordable pricing. Extremely challenging to get a reservation but the often have walk ins due to cancellations late during the lunch shift.
Zenit in Ostia by the beach. I got married here :) You can go during the day for the beach club and stay for lunch and/or dinner.
I'm assuming you've read on the basics (Colosseum, etc), here are some other options:
Giardino degli Aranci, a romantic and beautiful orange trees garden overlooking the city. If you're a 14-15 years old kid in Rome, you're taking 2-3 buses to take your date here to see the sunset :)
Castel Gandolfo is a city overlooking the lake in the Roman castles right outside of the city. The link takes you to a nice restaurant with a big balcony. Castel Gandolfo, Albano, Ariccia, Marino, and Genzano are the famous towns in that are if you want to do a day roadtrip.
Villa Borghese is right by Piazza del Popolo but not everyone goes up to explore it. You can take a long walk through the whole park and visit Galleria Borghese, one of the best art galleries in Rome.