Italy isn't too large: it is under thousand miles long, so that you can, if you want, drive from one end to the other in one, very long, day. Accordingly, the climate varies only mildly from the "cold" North in Milano to the warmth of Sicilia. The winters will be cool and wet anywhere you go, but you"ll see no snow in Messina. The summers will always be hot. What changes is the onset of the warm season - february has wonderful days in Sicily, while it can still be cold in Milan" may.
For most of Italy, may and september are ideal choices for a trip. Which one to choose depends mostly on your favorite occupation and the foods you enjoy. May brings artichokes, roses, strawberries, apricots and a flurry of fish and of fresh fruit. September is the time for the first delicacies of fall, as blackberries, pears and apples, chestnuts and so on. The climate is warm and mostly dry in both cases, and people will either be dreaming of summer vacation in may, or just come back from it in september.
If I were you, I would avoid the summer. Not that it's too hot: temperatures rarely climb into the nineties and it is newsworthy to find oneself in the one-hundreds. It's more that Italians are quite not there in the summer, and one finds that all tourist attractions are desolate and frequented only by tour buses and their lot, and by pickpockets on the prowl.
Things are changing in Italy, but as I grew up, you wouldn't be able to find food in august. Period. The newspapers would publish a list of the stores that were forced to stay open, and most of them would nonetheless post a sign claiming a sudden death in the family forced them to shut for the day, thirty days in a row.
Off-season can have its charms, as well. Venice, for instance, is never as wonderful as late in the winter, when the fog of the Po valley owns the city, and the only thing you hear all day is your own and other people"s heels clacking on the cobblestone pavement. Or how about the ski resorts in the Alps? Now, could Cortina be more spectacular than after the first snow falls?Next: Where should you go?