Grom on via Fillungo has amazing gelato, which one should consume as much as possible of while in Italy.
Rent a bike and cycle leisurely around the circular Renaissance walls surrounding the city. It is the best way to admire the town from above (and catch a breeze when it gets too hot).
Visit during the summer months and attend the festival, which has seen the likes of Nile Rogers, The National, Lionel Richie, John Legend, and Bob Dylan playing outdoors in the beautiful Piazza Napolene.
Along the wall there is a studio where opera singers practice. During the summer when the windows are open, the sounds stream out onto the roads around and it's magical to experience.