See details here
You can buy train tickets at the Tokaj train station Mo-Su between 05:30-18:40.
Direct IC trains to Keleti leave from Tokaj at 7:48, 9:48, 11:48, 13:48, 15:48 és 17:48. The train takes you to Budapest-Keleti in 2 hours and 42 minutes.
The ticket costs 4500 HUF, student ticket is 2500 HUF.
Direct IC trains to Nyugati leave from Tokaj at 7:04, 10:08, 12:08, 14:08, 16:08 és 18:08. The train takes you to Budapest-Nyugati in 3.5 hours, but you can also get off at Ferihegy, Kőbánya-Kispest and Zugló stations.
The ticket to Budapest-Nyugati costs 5800 HUF, 3300 HUF.
See the To Budapest by train section above.
These are the options to get to the Airport from the city center:
... BKK Public transport shuttle bus no.100E (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.)
Direct IC trains to Nyugati leave from Tokaj at 7:04, 10:08, 12:08, 14:08, 16:08 és 18:08 an take you to FERIHEGY train station in 3 hours.
The ticket costs 4800 HUF, student ticket is 3300 HUF.
BKK Public transport shuttle bus no.100E will take you to the Terminal 2 in 15-20 minutes (including the short walk to and from the bus). You'll need 1 ticket of 350 HUF. Ticket can be bought from the purple coloured vending machines in the bus stop.