Our response to the question "when do I go to Italy?" is "as soon as possible"! You can visit at any time of the year, depending on your tolerance to the weather. The only period I would tend to avoid is from mid-August to the end of August when Italians traditionally head to the beaches for their annual holiday and shops and some attractions tend to shut for those two weeks [it centres around the 15 August or "Ferragosto"]. Additionally, travel to and from Italy tends to be more expensive during this time since some Italians will go abroad during this time.
If you are free to travel at any time of the year, consider May or June when the weather is still somewhat moderate and the intense summer heat has not arrived. Likewise, remember that it can be quite cold in the hills and mountains of Le Marche and Abruzzo in winter, particularly if you venture inland. They will regularly get snow in the winter on ground above 500 metres or thereabouts. While the country does not grind to a halt, it is worth hiring or obtaining a set of chains if you are driving.
September and October are recommended months for a variety of reasons. The schools have returned, so you don't have to queue for major attractions. While a few tourist attractions finish at the end of August, most remain open through September and some continue beyond that. Weather is usually settled in September with highs of 20-25 celcius. Even October can be warm and pleasant although the natives often wrap up when it falls below 20C!