***Updated June 2023***
With the return of visitors to Rome, people are asking me for advice on where to eat. I tend to use Google reviews to look for places to eat. I try to avoid those that have under 4.5 stars.
First, my recommendations out of the famous places:
Pierluigi, Piazza de Ricci 144 (downtown Rome): it is on its own piazza and you can enjoy the people watching. It is a seafood restaurant, but it has the most delicious vegetarian pasta and tiramisu. You will need to make a reservation.
Colline Emiliane, Via degli Avignonesi 22 (near Piazza Barbarini): This place makes delicious food. It is the cuisine of Emilia-Romagna (Bologna, Modena, Parma, etc.). Small and with no outdoor seating. But, they have a window where you can watch the pasta being made fresh every day. You have to call to make a reservation.
Tonnorello (several locations in Trastevere): I thought it was a zoo eating there, way overcrowded and the same food you can get most places. If you go at 12:30 pm, you can probably get in to one of the locations.
La Matriciana (across from the opera house, near the Termini train station), Via del Viminale, 44: Classic place from 1870.
Then the less famous places:
Fuoco Lento, Via Flavia 63/65 (in Ludovisi/Sallustiano): Old school waiters, outdoor seating as well, never had anything bad here. It’s my “go-to” place. Open on Sunday night as well. Outside the touristy area so much easier to get a table.
Da Bucatino, Via Luca della Robbia 84/86 (in Testaccio): outside the tourist areas, has outdoor seating, and is an old school place.
Tratteria Valentino (not to be confused with Trattoria da Valentino on Via Cavour, which is also fine.), Via del Boschetto 37 (in Monti): great local place on a side street in Monti. Near Quirinale palace. Hidden in plain sight because they kept the old facade from when the space was a ice shop.
And the not famous:
Osteria del Rione, Via Basento 20 (near Villa Borghese): Just north of the Via Veneto neighborhood, this place is a real local place located in a basement. There is almost no other business on the street, an extremely limited menu (basically what Bruno, the owner, tells you he has), and a set menu for 10 euro.
Il Simposio wine restaurant, corner of Piazza Cavour, in Prati, but just across the river: They serve chocolate as a palate cleanser to desert. They also make good food and can also make all the old fashioned dishes. Actually, all restaurants can actually make the old fashioned dishes.
MiVa, Via Ezio 23: In Prati, away from the hordes. Modern Italian-American (in the sense that it’s like a place you would find in the USA) but of course does the classics as well. I like that they have chicken breast and avocado. Their semifreddi is excellent as is their apple pie.
Ristorante Pinseria Da Massi, Via della Scala 34: At the end or beginning of Trastevere. This is one of many good classic eateries in Trastevere. This places makes possibly the best spaghetti carbonara in existence… also, you can get fresh white (nothing but oil and salt) pizza fresh from the oven as your “bread.” Yum.
PRATI Rione Gastronomico, Piazza Giuseppe Mazzini 5: It’s far from the tourists but it’s a large space with plenty of room between tables, and the tiramisu presentation is a form for entertainment.
Forno Monteforte, Via del Pellegrino 129: Cafe, bakery, and oh so cute. Open all day and in a cute street. This is north of Campo di’ Fiore, and a whole world away. Grab breakfast, or sandwiches, or cocktails.
L’officina della pizza, Via Cicerone 22: One of the best places for pizza al taglio (by the slice). Located near a bus stop near Piazza Cavour, this place is open from around 10 (different on weekends) until 11 pm, or so. The most abundant variety, and homemade potato chips, sandwiches, and deep fried suppli (the Roman arancini), is around 1 pm. This is a must visit from many of my guests. The mushroom pizza and the fresh zucchini with stracciatella (the soft stringy creamy part of a mozzarella) are popular, but I like the ham and cheese.
Lastly, I recommend one place outside Rome (near the airport) Osteria dell’elefante, which I wrote about before.
very nice and noisy bakery to buy warm pizza alla pala, bread, cakes, biscuits … it’s so so so so famous. It does have lots of stuff and they are accustomed to tourists so don’t even try to speak Italian in there. 
… It is the organic store for Rome. 
Salvo patria, 
La condessa, Calle 85 (and other location): I hoped it would be great. They have fancy beer… But, the Florentine salad was a revelation. The lettuce was delicious. Completely undressed.