I'd second the Cinque Terre and Amalfi Coast suggestions if you've never been. Both are very popular so can obviously get extremely busy, but May is probably going to offer the best compromise between weather and crowds so if you're ever going to go to those places, that'd be the best time.
If you have a car at any point and are in the area, some of the mountain drives through Tuscany are quite nice. Not essential by any means, but there are countless cool little towns with very few tourists that you can travel through. It offers a nice counterpoint to the busier areas. The countryside in the triangular area between Florence, Siena, and Arezzo is gorgeous. The area north of Arezzo is quite nice too.
The mountain areas in the north are stunning, but might be a bit out of the way depending on where else you're going. The whole Bolzano/South Tyrol area is absolutely amazing; almost indescribably beautiful, though it does feel more like Austria than Italy. Further south, Lake Como and Lake Garda are pretty impressive and have some cool small towns around them.
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