We spent a week in Rome, a week in Florence and 3 nights in Venice in the spring of 2013, so your post brought back some nice memories! We travel a fair bit and usually use airbnb, but for some reason used another service for the Italy trip:
http://www.cross-pollinate.com/ You might find better deals on airbnb, but on the other hand, you are just booking now so some of the best deals may be gone. We usually look for places to stay that are just a bit way from the main tourist areas and if possible near a university. For example, in Rome, we stayed near via Tibertina between the train station Termini and a large cemetery, below the university of Rome. In Florence, we stayed near Piazza Santo Spirito, across the Arno from most of the sites. In Venice, however, we got a place on one of the main islands back from the main canal. We didn't try to save money by staying on the mainland and visiting Venice during the day, rather staying for a shorter time and having that immersive experience of Venice in all hours. We usually stay in apartments because then we don't have to go out for breakfast if we don't want to and we can keep water bottles cold and it's fun to go to the local markets--in addition to saving some money.
That said, eating in Venice was, we felt, expensive for the quality. We had the most fun eating at cicchetti bars. Cicchetti are like tapas and the are served in bars usually between 5 and 8 pm. In Rome, as another poster mentioned, the long pizzas where they cut what you want are fun, and the neighborhood where we stayed had some relatively inexpensive places to eat. In Florence, we ate at the central market, amazing food if you are an adventurous eater. It used to be open only until 2 but now it seems to have undergone renovations and is open much later. Not sure if the prices are higher and the food is still as good although we've usually had good luck finding good, inexpensive places near the markets. We also found some nice places near Santo Spirito.
We usually put a bottle of water and some fruit in a daypack so we don't stop to eat something just because we get hungry. Unfortunately many cultural sites are near pricey, touristy neighborhoods. We try to plan where we are going to eat before we set off for the day to avoid eating expensive, bad food. We don't necessarily pick the exact restaurant in advance, usually the area (a piazza, a street, etc.) and we usually plan on a "real" restaurant a couple of times a week and the rest of the time in bars or street food or at home. We filled up quite a few times at aperitivo time (around 7-9 pm) in Florence where bars had tapas, some substantial.
