k this thread is racist
k, like, there's no particular brand for what you've mentioned that you should get; the thing is, in italy food it's sort of culture, therefore the average level is pretty high, especially for the everyday things such as olive oil, tomato sauces and so on. so for olive oil just get the extravergine one, that is the best and probably the only one we eat here, as for the brand even those at the discount should be good enough tbh, maybe bertoli should be a good choice for you. as for cheese and "affettati" it depends on what you're looking for, i'm no particular lover of cheese but meat is my jam; honestly salame and pancetta aren't seen as fine cold meat, they're very cheap compared to the rest, and, tho i like them, i'd suggest you try something like bresaola (expensive indeed, but it's beef and it has very low percentage of fat) and prosciutto crudo (now you can buy the boxed one which is a bit lower than average or get it directly from the butcher, wether is good or not depends where he takes it from, if he just started slicing it it won't be extremely good (actually is shit almost), if it's the end of the prosciutto and it's close to the bone, it will taste sweet and my goodness :Q ). prosciutto cotto ("di parma" is the best), mortadella and coppa for cheap but very tasty affettati.
i don't know if you like vinegar too, cause we do, and if you'd like to buy some take the aceto balsamico (possibly from modena), love it but just few drops (it's very strong flavor), and only with salad.
and one thing for god's sake, there's one thing you cannot forget about italian food: fucking. red. wine. do NOT, i repeat DO NOT leave italy without tasting red wine (problem is we have hundreds of red wines, brands and vineyards and shit, but the three i would suggest are montepulciano d'abruzzo (very cheap but quite nice taste, especially for the price) montefalco rosso (do not mistake it with sagrantino di montefalco, one of the best wine in the world and as such very expensive lol ) and chianti ofc), or if you like light and quite sweet wine refosco dal penducolo rosso.)
if you need anything just ask, but i believe other italians should know things better than me (i don't usually do the grocery shopping). the last thing, about olives; honestly i prefer the greek ones but that's my personal preference. oh and ofc pizza, try the one with pomodorini (cherry tomatoes, they're smaller but much more tastier and sweet tomatoes), it's my fav these days.