[This is what I remember from studying P&H]

In Roman times it was seen as a luxury to have home cooked meals as only the elite could afford it. The general population would buy fast food from a thermopolium, in which food could range from breads, fruits, cheese, olives and alcohol. It is seen through a study by Estelle Lazer, that the teeth were particularly damaged due to the minute stones that would intermingle with the dough before cooking. Her study further shows that the Pompeians were quite well nourished and ate a lot of the right types of food.

A posh Roman would "eat out" at someone else's place - Roman dinner parties were seriously big deals in sociological terms, it's important to bear in mind that 'houses' aren't just abodes in Roman society, they're combined familial/political headquarters, meeting place and temple (with shrines to household gods and such). Putting on dinners for prominent guests (especially one's own patrons), and your own clients, survives as having been very important in the Roman attempts to climb the greasy pole.

For an especially funny account of a Roman would-be client trying to impress a patron with their dinner party skills, check out Horace's Satire 2.8, in which one Nasidienus Rufus makes a dog's breakfast of trying to impress Maecenas with his elegance and affluence.

Recent thread on the topic has some answers for you (but not all).