How to reach Szeged:By air, from Budapest Airport (Terminal 2):
Take a bus #200E from Terminal 2 to "Ferihegy vasútállomás" or "Kőbánya-Kispest" (other end stop). The timetable of this bus can be found here. Buses usually depart every twelve minutes. You can buy bus ticket (320 HUF) at the terminal bus stop (from a vending machine; coins only) or from the bus driver (coins only; on this line).
The bus stops at "Ferihegy vasútállomás", which is close to Terminal 1 (currently closed terminal) and to the railway stop called "Ferihegy". There you can buy a train ticket to Szeged from a vending machine, or on the train from the ticket inspetor (with an additional 2600 HUF). It is recommended to change to trains at this stop because this stop is closer to Terminal 2 and it is very easy to find the train stop.
Or, at the other end stop of the bus line, there is a railway station called "Kőbánya-Kispest". There you see a covered pedestrian bridge over the railway tracks, and there is a train ticket office on the bridge.
By train, from Budapest:
Szeged is the last stop on the train route. From Budapest trains leave from Nyugati Pályaudvar (Nyugati or West Railway Station) and they also stop at Kőbánya-Kispest railway station and at Ferihegy railway stop.
The official, online timetable of trains and ticket prices can be found here:
from Budapest Nyugati to Szeged and from Kőbánya-Kispest to Szeged and from Ferihegy to Szeged.
Please dont forget to set the date. The (one-way) train ticket costs around 3410 HUF (about 12 EUR). It is a good idea to ask for a seat reservation ("Helyjegy"). If you do not have a ticket you'll have to pay an additional 2600 HUF on the train.
Take the autostrada E75 (M5); it has two exits for Szeged (either can be used). Or use Google Maps to get driving directions.