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)
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.
Take the train from some other...
Take the train from some other place such as Budapest. Bring your own sludge-free coffee and walnut pastries