These are the options to get to the city center:
... BKK Public transport shuttle bus (900 HUF, tickets can be bought from purple coloured vending machines in the bus stop or at BKK Customer Center at the Airport Terminal.)
... Flibco Shuttle to the Keleti Station (from 9€, you might consider booking in advance) or
... MiniBUD Shuttle door-to-door service (from 16€, you might consider booking in advance) or
... FŐTAXI to the Keleti Station (about 6600 to 8000 HUF, service desk at the entrance of terminal. Please make sure you take no other cab than FŐTAXI, beacuse others may cheat you by taking you at a much higher rate.)
From Keleti Railway Station direct IC intercity trains to Tokaj leave at 7:30, 9:30, 11:30, 13:30, 15:30, 17:30. The journey takes 2.5 hours.
From Nyugati Railway Station direct IC intercity trains to Tokaj leave at 6:23, 8:23, 10:23, 12:23 and 14:23. The journey takes 3.5 hours.
Take BKK Public transport bus no. 200E to FERIHEGY VASUTALLOMAS. The journey will take about 15-20 minutes including the short walks to and from the bus stops. You will need 1 single ticket which costs 350 HUF. You can buy the ticket from the purple vending machines at the bus stop or at BKK Customer Center at the Airport Terminal.
direct IC trains:
Direct IC intercity trains from Nyugati to Tokaj leave at 6:43, 8:43, 10:43, 12:43 and 14:43. The journey takes 3 hours.
Click here to check the timetable and buy your ticket online or via the MÁV app on your smartphone.
The ticket costs 5300 HUF. With a valid student ID card** it costs 3000 HUF.
*PLEASE NOTE: If you have a Student Identity Card issued in the EU containing your photo and stating the name of your school, you are eligible for a 50% discount, and you can buy a student ticket.
...You can buy them online or via the MÁV app on your smartphone. If you buy your ticket online, you will get a 10% discount.
...At the train stations, either at the cashier or from a vending machine, or
...At Keleti, next to the train track No. 9, there is a travel agency, where you can buy in-land as well as international tickets, and they speak English.
From the Tokaj train station a 10-minute walk will get you to the CSI Main Building (i.e. the Tokaji Ferenc Secondary School).