The central bus station is on Buza ter (next to a big market) and is just a couple of minutes off the main street Szechenyi ut. Buses from here to many destinations including Eger (1 hour 30 minutes), Aggtelek (2 hours), Gyongyos (2 hours).
There's a ticket office at the station, but you can just pay the driver when you get on the bus.
The train station (Tiszai) is a 10 minute tram ride from the centre -- there's a kiosk outside the train station where you can get a tram ticket -- take tram nos: 1 or 2, both run down the city's main street Szechenyi ut.
24-hour luggage storage in the basement.
Frequent trains to Budapest (journey time approx 2 hours 30 minutes)
Take the train from some other place such as Budapest. Bring your own sludge-free coffee and walnut pastries