According to my best foodie friends, here are some of the most amazing places to eat at in Rome: 

1. Roscioli Salumeria con Cucina. Get the salumi and carbonara.
2. Try an Espresso Granita. Lots of sugar. Good ones near the Pantheon. Perhaps try Tazza d Oro.
3. Antico Forno Rocioli. Bakery and Roman pizza shop easy to access. Amazing bread.
4. Bonci Pizzarium. Close to the Vatican. 
5. Mordi & Vai (Box 15) at Testaccio Market. Get the brisket sandwich (Alesso di scottona). Also try the marinated artichokes. 
6. Piatto Romano. Nearby the Testaccio market.