You can reach Szeged and Pécs very easily by Intercity (IC) trains. The journey lasts just a bit more than two hours. The trains to Szeged depart from Nyugati pályaudvar (Western Railway Station), to Pécs from Déli pályaudvar (Southern Railway Station). You can check out the time tables at www.elvira.hu

If you come from the airport it is best to take the Airport Minibus to the railway station. (The taxis waiting at the airport are much more expensive.) From the city you can reach Nyugati pályaudvar by the blue metro (subway), Déli pályaudvar by the red metro.

Please, do not forget that you need to buy a supplement for the IC trains.