Grocery stores in Rome are small, but, you will probably be able to find everything you want or think you want. For some speciality stuff, you will need to go to the international stores as the basic chain stores have very little “international” food. While the chain stores are convenient, I actually prefer the little mom and pop stores, but they are dying away.
Pam, Coop, Conad, and Carrefour are the main chain stores in the center of Rome. Carrefour is tiny here, unlike in some places where they are megastores (like in France).
For the Wegmans/Whole Foods type stores, there is DOC (by Coop) and NaturSi (some are quite basic and others have a wider variety like the one on Piazza Farnese, which also has a cafe).
For the bargain chains, there is Todis, Lidl, Metro, and some that I can’t recall the name of, maybe Tiger?
The largest combined “super-Target” type stores I’ve seen here are Conad and Elite.
Most of the local “rione” markets will have everything for basic shopping but markets tend to close by 2 pm. Many of the specialty stores like delicatessens and bakeries have started carrying other products like milk so that you can do much of your grocery shopping there. I can find almost everything I need for grocery shopping at the farmers market on Via San Teodoro, but it will all be made locally, so no imported goods.
The hardest products for foreigners and expats to find in Rome is cilantro and corn tortillas. I notice a trend in “Mexican” food so these products should be easier and easier to find over the years. Again, finding TexMex is a subject that continually follows me around the world. Gringos are obsessed with it.
For international products, Castroni is a shopping emporium with an Aladdin’s den of dry goods, but also refrigerated goods. There are many Asian shops that carry goods from all over the world, including some fresh vegetables.
For those looking for vegetarian foods or gluten free foods, all these are now easily found in most grocery stores. “Exotic” foods require a bit more hunting. Also, sometimes, the store may not have a whole lot of the one product that you want. I consider it a victory when I can buy more than one bottle of my favorite type of milk.