Eger is mostly known for its castle, thermal baths, historical buildings (including the northernmost Turkish minaret) and for the best Hungarian red wines.
Majority of travellers come here for a single day, not staying overnight. Ukranians (and Russians) are frequent among visitors to Castle, Italians go next. Around the town, many Deutsch-speaking travellers can be seen (Austria?).
Train from Budapest. Car from Miskolc through the windy roads of the Bükk mountains.
It's small and best walked on foot. However, the town has a taxi service especially useful if you arrive by train (phones: 555-555; 333-333). Also, if you go for some outdoor activities near the town there're some buses connecting the town with the suburbs.
Definitely good place for wine shopping. You can buy directly from the makers.
Plenty of restaurants all over the town.
Visit some csárdas in Valley of Beautiful Women (Szépasszony-völgy). Some of them are contanly overcrowded as tourists are brought there by buses of partner agencies.
One of the specialities are pancake balls.
Try Fehér Szarvas Vadásztanya (White deer hunters Inn) that offers excellent game dishes. Not the cheapest in town, but the meals are excellent and the athmosphere is unique. The restaurant is located downstairs next to Hotel Park.
Wine. Eger is the centre of one of the top wine regions in Europe. Egri Bikavér (Bull's Blood) is most famous cuvee here, but there are several others definitely worth a sip.
Several camp sites. Plenty of small and larger hotels and private rooms all over the town. You will not have a problem finding accommodation there.