Food shopping in rome
Definitely check out VOLPETTI in Testaccio, an intense, one-room, packed-to-the-gills place where most of the goodies are behind the counter - everything I tried or recognized from previous experience was amazing. The focus is on cheese, meat, fresh pasta, baked goods, and other fresh foods, but I remember them having an excellent olive oil selection as well. Definitely stop by Volpetti's tavola calda, Volpetti Piu, around the corner - it's cafeteria-style and the quality is terrific... I still drool thinking about my plate of artichoke hearts, cooked generously in olive oil and garlic and herbs until parts were brown and almost crackly, with other parts tender, totally alchemic and delicious. At Volpetti Piu they offer five or six different quality olive oils to drizzle on your food - fun to taste each.
Another food shop we loved is INNOCENZA on Via Natale just off Piazza San Cosimato, a short walk across the river from the Testaccio market. This place focuses more on dry goods - sweets, jams, antipasti, large paper sacks filled with various beans and grains, pastas.... Everything is very well-priced and remarkably good. We especially enjoyed the a wide selection of Domori varietal chocolates for less than 3 euros each.
Here are a couple lists of markets around town:
For non-touristy shopping I like the large covered market at Vittorio Emanuele.
Campo dei Fiori market is certainly colorful, scenic and convenient, but definitely does not have the best prices - actually the prices are among the highest in the whole city. Best selection is also arguable - yes, you can find things there that are hard to find elsewhere (passion fruit, parsnip, etc.) but when it comes to local produce, there are other markets that have better quality (e.g. the already mentioned Testaccio and Trionfale markets, and there are many more).
Beside the aformentioned places all very good I would suggest to visit Macelleria Feroci (just off the pantheon) & Antica Norcineria Viola (campo de fiori) for dry sausages and cured meats, Franchi & Castroni (one next to the other on via cola di rienzo) for cheese and olio nuovo (are you looking for any specific area or DOP?), while for a bit of everything you could try Alimentari Elite on Via Cavour 230/236 or the big indoor market called Mercato Doria a short walk from the Vatican Museums on Via Andrea Doria.
Hope this helps:-)
Anyplace EXCEPT the supermarkets (but not across the board) and the Campo de' Fiori market is probably fine. The Roscioli shop on via dei Giubbonari, near Campo de' Fiori, has everything top quality.
The Testaccio quarter and the Volpetti shop in particular are excellent, but really the products you mention are readily available throughout central Rome. The shops in via della Croce are good. The Palatium wine bar in via Frattina has products of Lazio.