When is the best time to travel to Italy in the summer?

Avoid August if at all possible: it's often extremely hot and that's when most Italians take their summer holiday. July can also be quite hot and touristy places are often quite busy. September is generally good, since the weather is still quite warm and the staff in places like hotels and restaurants have started to recover their good humour after the peak tourist season. June is also good because the weather is usually not all that hot and the countryside has not yet really gone all dry and dusty as it will in August. As for what sort of things you should do while here, that really depends on what sort of things interest you. You could do what Italians do and go lie on a beach for two weeks or you could spend all your time traipsing around museums. Maybe you should ask a question about what to do and say what sort of things you like doing on holiday and in what part of the country you'll be in.