Someone asked in one of the travel forums about how to eat cheaply in Italy, and whether it’s possible to take a "doggie bag" from a restaurant.
To answer the second question first: I’ve only once taken away the remains of a meal from a restaurant in Italy (a steak that was larger than anticipated). Italian restaurant servings are of a reasonable size, so usually if you clean the plates on a first course (carbos: pasta or rice) and second course (protein + veg), you will be comfortably full, but not bursting. But if you did have anything leftover, I’m sure they wouldn’t mind wrapping it for you.
The restaurants that also (or exclusively) do takeaway – look for the sign "asporto" - often serve Chinese food, kebab, or pizza. Best of all, however, are rosticcerie, which make rotisserie-grilled chicken (which you can buy whole or in parts), roasted potatoes, and a large or small variety of other dishes. They are better equipped with take-away containers, and will give you napkins and plasticware as well.
There are other ways to eat cheaply. Most supermarkets and some smaller stores have a prepared-foods counter with both hot and cold food ready to go. You could also buy fresh bread, cheese, prosciutto, salame, etc. and make your own sandwiches. Buy some olives, pickled onions, and other goodies to round out your meal. You can buy fresh focaccia and pizza at bread bakeries (panetterie); most will heat it up for you (in a real oven, not a microwave, so be prepared to wait). If you buy rolls for sandwiches, you can ask the baker to slice them open for you, ready to receive the sliced meats from the butcher next door.