Location is important: city vs rural. I will assume an urban environment.

For the average plebeian in the middle of the 1st century CE, the diet is dominated by bread (either emmer or wheat loaf), olive oil, fish sauce (garum), vegetables, and wine. The wine was typically what we moderns would consider extremely cheap and foul, more on the scale towards vinegar than modern wine. Meat was not every day but also not as rare as it is sometimes made out to be. More on meat below.

Many of Rome's hundreds of thousands lived in insulae, apartment buildings, and elaborate cooking ventures were not the best idea. Still, small open hearths or little ceramic stoves were very common and frequently caused fires. Some storage of food stuffs was possible, unrefrigerated of course, but not strictly necessary. All the daily needs of an urban dweller were immediately available every day from local sellers down on the street. There were large "farmers' market" style areas throughout the city, like the meat markets and the vegetable markets, and then from there foods were distributed (and the price hiked) to smaller neighborhood places. The situation was virtually identical to the little corner markets you can find in any dense urban environment today, from modern Rome to Paris to Manhattan.

Bread was ubiquitous, and bakeries were everywhere. The white breads we know today, like sourdough, were expensive; the lower classes got cheaper, "darker" breads. Olive oil too was everywhere, for everyone, with variations in quality. Olive oil was a major source of calories for virtually all denizens of the Mediterranean basin. It was absolutely essential and would have been present at virtually any meal for any person, rich and poor. I mentioned fish sauce, which was also very important. It was the Roman condiment par excellence, equivalent to the roles of ketchup, mayonnaise, mustard, and soy sauce all in one product. There were huge variations in quality in price, from the very fancy stuff all the way down to the very cheap stuff. I once spent an entire semester at university studying Roman fish sauce and I'd like to forget it. Also, most westerners would find an average Roman fish sauce to be incredibly vile and nearly inedible.

Vegetables were everywhere for everyone. Cabbage was common, as was garlic and an ancient form of broccoli. Onions, turnips, beets, carrots, radishes, peas and beans of many sorts, lentils, pulses, leafy greens like kale, chard, lettuce. Spinach was not around back then, nor was corn. From the New World come potatoes, tomatoes, and many peppers (I'm not sure which are from the Americas and which from East Asia), which are now essentials to Italian cooking. They did not exist in the ancient Med world.

Meat: the common line is that the plebs never ate meat except at public festivals (when it was provided as part of the party), but a lot of garbage heap evidence is these days seen to refute that notion. A Roman pleb had about the same purchasing power as an average American History grad student (which is to say, not much), but they could get meat on their own when they wanted it, maybe just not fillet mignon every day. The common meats of the Roman world: beef, of course, including the good stuff like tongue and liver and such; pork; chickens and other birds, like pheasant and duck; venison; goat and mutton seems to have been less common; and Romans, like most ancient societies, did not eat horse flesh except in dire circumstances. Fish availability and price was regional, but the ancient seas abounded in fish resources. In Diocletian's Edit of Prices (late Empire, not early), sea fish seem to be expensive, twice as much as an equivalent weight of pork, but river fish are half as much as pork. Those are the maximums, anyway. I don't actually know if Romans drank milk, but butter existed, and was fairly expensive.

Someone recently asked me if pasta is ancient, and I don't have an answer for that. The Romans had lagana (perhaps ancestor to modern lasagna), but it was really just sheets of dough. I think modern pasta as we imagine (linguine, rotini, farfalle, etc) are from the Medieval period. I admit my pasta knowledge is substandard for this subreddit.