There is only one airport in Budapest.
The organizers will try to arrange for transportation from the airport to Szeged for flights arriving during the day
Szeged is the last stop on the train route. From Budapest trains leave from Nyugati Pályaudvar (Nyugati or West Railway Station). If you need to take a train from the airport there are two possibilities:
I. The Budapest-Szeged trains stop at Terminal 1 of the airport. Now all flights arrive to Terminal 2, from where you need to take a taxi (simpler) or a bus #200E. Buses depart about every twelve minutes, here is the timetable of the bus. You start with the bus at Liszt Ferenc Airport Terminal 2, and you'll have to get off the bus after about 12 minutes at Liszt Ferenc Airport Terminal 1. There you can embark the trains to Szeged. Unfortunately, there is no possibility to buy a train ticket at the airport or at the train stop, so in that case you'll have to buy it on the train itself (additional 2600 Ft).
II. The Budapest-Szeged trains stop also at Kőbánya-Kispest station which can be reached from the airport with the same #200E bus in about 22 minutes. At Kőbánya-Kispest you see a covered pedestrian bridge over the railway tracks, and there is a train ticket office on the bridge. The train ticket costs 3410 Ft (about 12 Euros), ask for a seat reservation ("Helyjegy"). If you do not have a ticket you'll have to pay an additional 2600 Ft on the train.
The online timetable can be found here:
from Budapest Nyugati to Szeged and
from Kőbánya-Kispest to Szeged.
Take the autostrada E75 (M5); it has two exits for Szeged (either can be used). Or use Google Maps to get driving directions.