Perugia

Perugia needs 2 days: the Etruscan-medieval historic center on a hilltop, with the 13th-century Fontana Maggiore in its main square, on one day, and a trip to Lake Trasimeno, central Italy's largest lake, on the other. April-October is best. It's inexpensive by Italian standards. It's the capital of Umbria and home to one of Italy's oldest universities for foreigners, giving it a lively student scene, and every July it hosts Umbria Jazz, one of Europe's most important jazz festivals.