The cheapest option is to travel in the city is to use the public transport system starting with the no. 93 (Airport Bus) which will take you to Kőbánya-Kispest Metro station (M3) and on to every other line. This could cost as little as 300Ft per person.
On the BKV web page you can found a bus called "Repter-busz" that serves this route. What does "Repter-busz" mean? "reptér" means airport, "busz" means bus. You can use the airport bus with BKV ticket. If you come out from the arriving level, you will find the bus stop right there, it's hard to miss! It takes about 22 minutes with the bus (if there is no traffic jam) to get to Köbánya Kispest where you can catch the subway num 3 (M3).
At the airport (on departing level) there is a shop where you can buy ticket. You can get one on the buss as well! If you happen to buy your ticket from the driver, you have to validate it as if it was a pre- purchased ticket. Most tourists get caught by controllers on Repter-busz because they buy their tickets from the driver and believe that it is already a valid ticket that they get from the driver - but that's not!
A recommendation: don't forget to validate another ticket when you change, unless you have bought a daily, 3-days, weekly or 2-weeks ticket.
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