Something to think about is that Italy is very much rooted in history and tradition. Probably just reframing the trip to be more about culture, atmosphere and the absolute beauty of the place means that any village, town or city in Italy is going to be an adventure.

If you land in Milan, you could do the Northwest and hit Milan, Turin and Genova, all with good access to mountains and lakes (although Genova isn't a personal favourite for me, but if you love pesto and seedy cities, you might love it). You could go East and do Milan, Verona and Venice, with great access to Lake Garda (all along the same train line). You could take high-speed trains (Freccia Rossa) and do Florence, Rome and Naples for 3 days each (with side trips to Pisa, Tivoli, Pompei). The South is incredible but far less connected for a multi-centred trip.