Thank you all again for all your wonderful advice and suggestions!
In about an hour we'll be leaving Rome for Florence, and I just wanted to come by quickly and share some of our experience. Overall, we had fantastic food here and have zero complaints. After the first day, I realized I was going to have to cut back on some of our restaurant plans simply because it was too much food.
Armando al Pantheon: We made reservations via email for our first night in Rome. And it was good that we did as probably two dozen people (both tourists and regulars) were turned away. We ordered carbonara, cacio e pepe, lamb, an artichoke, and house white wine. The bill was €58.
Service was kind of brisk but friendly. We tried to speak (terrible) Italian and they quickly switched to English. We thought the pasta was good, but the lamb was great. Really, really great.
Pizzarium: We went twice. =) The first time we got two suppli (one spaghetti, one sausage/pasta/artichoke), pizza rossa and pizza patate. Bill was something like €12. We ate the pizza first and loved it, but the suppli was out of this world. I can't figure out why I've never seen it in the States since fried balls of stuff is right up our alley.
The second time we got two suppli (classico & spaghetti), a pizza with broccoli/sausage/mozzarella and one with prosciutto and Gorgonzola. We loved the broccoli pizza but found the other okay. Our bill was €11.40.
Pizzeria da Remo: We ordered two pizza Margerhita, two suppli, and house white wine. The bill was around €18.
The pizza was great, and compared to PIzzarium the suppli was kind of meh. Service was brisk and I don't think anyone there spoke any English. We did okay with our terrible Italian. Regulars were greeted warmly, and the place was full of them. We were the only tourists.
Flavio al Velavevodetto: We ordered carbonara, ravioli, tripe, tiramisu, and house white wine. The bill was €48. We were able to call on Thursday to make a reservation for Friday night.
The pasta was fantastic. Utterly wonderful. The tripe was good really well and the flavor was great, but I think the texture was not really for me. We tried it at the suggestion of the server and are glad we did. The tiramisu was amazing, and even my husband liked it.
Service was really great. They took the time to stand and explain the menu in English. They made suggestions and were really helpful. We left a few Euro as a tip when we left.
We had gelato all over the city, and liked most of what we got. We managed to get to Palazzo del Freddo on half price night, which was a happy accident.