You can use the railway to travel from airport to Debrecen and also for traveling within Hungary or visiting the neighbor countries.
In case you have decided to travel by train from the airport to Debrecen, you need to take first a bus Nr. 200E. Its stop can be found on arrival level next to Terminal 2B. When you come out from the building, turn right and you will see the sign of a bus which shows the stop of it. Click here for Timetable of bus Nr. 200E
This bus takes you to a train stop called „Ferihegy vasútállomás”.
You need to buy ticket both for the bus and train. Bus ticket can be bought from the driver. The easiest way to buy a ticket for the train is buying it online. For timetables and online purchase click here.
The trip takes two and a half hour to Debrecen. When you arrive to Debrecen you can take bus, trolley and tram or catch a taxi depending on your destination.
About public transportation you can read more here: Public transportation in Debrecen.