Hi Elizabeth! A little-known place in Rome that I like for lunch is Antica Osteria da Giovanni, near the Vatican. It's on Via della Lungara 41A. Very simple traditional Roman food at honest prices. There are a few tables for outside dining and the rest inside. It gets crowded so you might want to try for lunch a bit earlier than the Italian time, say around noon or 12:30 instead of 1 pm or 1:30.
Another spot I like, which is similar, is Da Tonino on Via del Governo Vecchio 18/19. Definitely enjoy the house bread, a traditional Roman roll called a "rosetta" that is hollow in the middle.
Neither of the above places has a website! Very non-touristy, locals love. Neither are fancy. These are old-school trattorie. You won't be able to book ahead though. You have to just show up when you want to eat.
As for Pompeii, our company ItaliaTours has just launched a really amazing day trip that includes a wine tasting at a prize-winning vineyard and a farm-to-table lunch featuring local specialties. That one you can definitely book in advance, and it includes both the train down to Pompeii as well as bus transfers, and a guided tour of the archaeological site by a Pompeii born-and-bred archeologist, so it is truly all-inclusive. You can see more details at our website here: Pompeii from Rome with Wine Tasting and Lunch20. I'm copying my colleague @Danielle 14 who runs ItaliaTours; in case you have any additional questions, you can ask them right here!