You'll pay some 2,500 Ft ($13/£6.85) for a double room in a private home in Veszprém, and you can find a list of accommodations at www.veszpreminfo.hu. The room price usually does not include breakfast. You can book a private room through the tourist office.
Péter-Pál Panzió (tel. 8/832-8091; www.hotels.hu/peter_pal_panzio), is conveniently located on Dózsa György u. 3; it's a 5-minute walk from the center of town. Don't be put off by the grungy building facade. Inside are 12 tidy but very small rooms, all with twin beds, a shower-only bathroom, and a TV. Insist on a room in the rear of the building, as the pension sits close to the busy road. The rates is 8,900 ($48/£25) for a double. Breakfast is included and is served in the garden in summer. Call ahead for reservations.
Hotel Villa Medici (tel./fax 8/859-0070; www.villamedici.hu), at Kittenberger u. 11, is a modern, full-service hotel set in a small gorge on the edge of the city, next to Veszprém's zoo-park. There are 24 double rooms and two suites; each has a bathroom with shower and the usual hotel amenities. Rates are 24,300 Ft ($131/£69) for a double room and 32,900 Ft ($178/£94) for a suite. Breakfast is included. The hotel also features a sauna, a small indoor swimming pool, and a salon. Check the website for package deals. The reception staff speaks English and will make tour arrangements for you. Major credit cards are accepted. Take bus no. 4 from the train station to the stop in front of Veszprém Hotel, and then change to bus no. 3, 5, or 10. These buses will take you as far as the bridge overlooking the gorge. You can walk from there.