Taxi. Zóna Taxi +36-1 365-5555 has the right to take passengers from Ferihegy airport. Depending on your destination, a trip to Budapest costs between HUF 3900-5700 (EUR 17-27) - the fare is slightly more expensive if paid in Euros. Queue at the taxi stand to receive a written quote for your fare, then pay it when you arrive at your destination - this system is designed to eliminate unjustified price hikes. IMPORTANT: unless you have pre-ordered a taxi from a different company, do not accept offers from taxi drivers waiting in the terminal or near the terminal entrances. This is for your own safety. On your trip into town you might receive a business card from Zona Taxi quoting cheaper fares bound to Ferihegy airport (i.e. EUR 16 from Pest). Reserve a car by phone and quote the offer to save some money.
Minibus service. If you travel alone, consider the Airport Minibus service, a shared taxi operation that collects passengers going in the same direction and will take you to or from anywhere in Budapest for HUF 2990 per person, HUF 4990 for a round trip. Join the queue at the airport and you will be on your way in 15 minutes. For the trip back, call the center +36-1 296-8555 (at least 24 hours beforehand) and Airport Minibus will pick you up.
Bus. From either Terminal 1 or Terminal 2, Bus 200E takes you to to Kobánya-Kispest metro 3 station. Buses stop outside the airport terminals. The journey to the metro station takes approximately 26-30 minutes from Terminal 2 or 15-17 minutes from Terminal 1 if there is not too much traffic. Buses run every 8-20 minutes from 05AM to midnight. There is no night bus service between the airport and the city, but the last four departures of bus 200E are connected to the night buses 914 and 950 which replace the metro. Note that if there is a lot of traffic, the journey can take an extra 10-15 minutes. You can also take the 93 from terminal 1. It will bring you to the same metro station, but it's considerably slower.
If you arrive in terminal 2B, take note that the buses are to be taken from outside terminal 2A, which is 50 meters to the left after you exit. Before exiting terminal 2B, be sure to buy some bus tickets from the newspaper vendor.
Bus tickets are available in airport terminals for HUF 320 at the newspaper vendors, or HUF 400 if you purchase directly from the bus driver. Note that a single ticket is only valid on the bus. If you continue with the metro from Kobánya-Kispest to the central city you need to buy a new single ticket. You will be better off buying 2 tickets from the airport, one for the bus and one for the metro, as the automatic machine in the metro station only takes coins.
The bus ticket has to be validated inside the bus, while the metro ticket will be validated in the orange machine present at the entrance in the station.
Railway. There is a railway station connected to Terminal 1 via a covered bridge. Passengers from Terminal 2A or 2B can also reach it by taking the 200E bus to the stop called "Ferihegy vasútállomás" (10 min). Frequent suburban train services run westwards to Budapest Nyugati station via Kobánya-Kispest and eastwards to Vecsés, Üllo and Monor, and semi-fast services to Cegléd (with additional services to Abony and Szolnok during peak hours) which only stop in Ferihegy between Kobánya-Kispest and Monor. Note that although these trains are suburban trains, they are operated by MÁV (the Hungarian rail operator) and by no means included in the HÉV network. The stopping service to Monor (called Személy) normally has 2 trains/h while the Cegléd service (called Zónázó) has 1 train/h (2 during peak hours, 1 of which continues to Szolnok). The travel time to Budapest is 20 min with a Zónázó train and 23 min with a Személy train. A single ticket to Budapest costs 365 HUF and can be bought at the information desk at the airport or from the ticket vending machines at the station. Local transport passes valid for two weeks or longer can also be used. For timetable information, check the Hungarian Railways website. Use the words "Ferihegy" for the airport and "Nyugati" for the city center.
Long-distance trains called Intercity or Gyors also operate from Ferihegy to destinations like Kecskemét, Szeged, Püspökladány, Debrecen, Nyíregyháza, Miskolc and Záhony at the Ukrainan border. You may also encounter Intercity trains with Budapest-Keleti station as the destination, BUT UNDER NO CIRCUMSTANCES WHATSOEVER SHALL YOU EVEN THINK ABOUT TAKING THIS TRAIN IF YOU'RE GOING TO BUDAPEST. It will take you to Budapest, yes - in 5 h 45 min, through Szolnok and Debrecen and Miskolc and everything in between (to clarify, this is a ring line which starts at Nyugati and goes through Ferihegy and eastern Hungary before terminating at Keleti, and the other way round). Using long-distance trains to Nyugati is not recommended either, as this will require a seat reservation, and this might cost a lot.