There are many cafes and restaurants in the Historic Centre - almost on every street, including the areas around Corso Vannucci; Via dei Priori; Piazza IV Novembre and Via Ulisse Rocchi.
The prices are not too expensive - probably because it's a university town.
Restaurants are generally open between 12:00 - 14:30 and 19:00 - 23:00, with cafes open all day selling pizzas and sandwiches etc
There are some good Patisseries including Sandri on Corso Vannucci & Caffe Del Sopramuro on Piazzo Matteotti
There are no vegetarian cafes or restaurants that I could find (and I'm a vegetarian!).
I'm a vegetarian: Sono un vegetariano
I'm a vegan: Sono un vegetariano integrale.
(Note: Many Italians will think you eat fish even if you tell them you're a vegetarian so it is best to use the following)
I don't eat meat: Non mangio le carne.
I don't eat fish: Non mangio le pesce.
I don't eat eggs or dairy: Non mangio le uove e non bevo latte