Bunny is the house specialty. Try the sweet dessert wine. See Civita and then find your own private hilltown. Gubbio, Volterra, Cortona, and Arezzo are discovered but rarely visited. Virgin towns are simply pleased you dropped in.
Any guidebook lists the popular hill towns.
But if you want to dance at noon with a toothless lady while your pizza cooks, press a good luck coin into the moldy ceiling of an Etruscan wine cellar, or be introduced to a mediocre altarpiece as proudly as if it were a Michelangelo, stow your guidebook, buy the best local map you can find, and explore. Talk to travelers who have studied or lived in Italy. Leave the train lines. Take the bus, hitch, or rent a car for a few days. Ask locals for their favorites.