Veszprém does not have many dining options, but you should be able to find a satisfying meal at the following places:
Cserhát Étterem (tel. 8/842-5441), housed in the huge structure at Kossuth u. 6, is an old-style önkiszólgáló (self-service cafeteria). You'll find this very popular cafeteria behind some clothing stores; go up the winding staircase inside the building. Hearty traditional meals are available for less than 500 Ft ($2.70/£1.40). The menu changes daily; it's posted on a bulletin board on the wall inside the restaurant upstairs. It is open Monday through Friday from 10am to 6pm and Saturday 10am to 3pm.
For something more upscale, Villa Medici Étterem (owned by the same people who own the hotel), at Kittenberger u. 11 (tel. 8/859-0072), is the place. It's expensive for Hungary but worth it. Main courses here are between 3,250 Ft and 5,200 Ft ($18-$28/£9.25-£15). Villa Medici serves Hungarian/continental cuisine daily from noon to 11pm.