Taking the public transport seems to be the most reasonable option but of course you can also choose a taxi or minibus transfer to get to your destination. It's more expensive, though.
Bus & metro
Passengers are taken by a direct bus service from Ferihegy/Liszt Ferenc airport called Bus 200E to Köbánya-Kispest Metro Station.
By the 'blue’ (number 3) Metro you can reach Deák Ferenc Square in the City centre in 20 minutes.
Pay attention to the timetables if you arrive late in the evening, because the last metro runs at 23:15 from Kőbánya-Kispest, and if you miss it you must use buses of the night service (it's not that safe or comfortable) or rather a taxi.
From the Airport to the city centre by public transport: lines and times.
The public transport services’ map available here: http://www.bkk.hu/en/maps/
A single ticket costs 350 HUF, at the airport it is purchasable from the Custumer Service Point or a vending machine . You need to validate a ticket on every single vehicle you board to. (More ticket and pass: http://www.bkk.hu/en/prices/) More info about Budapest's public transport: https://www.esn.bme.hu/getting-around-budapest
There is just one contracted taxi company that can drive you from the airport. Here are the fixed fares, so don’t accept higher price: http://www.bud.hu/english/passengers/access_and_parking/by_taxi
Towards the aiport you can choose any company you’d like to.
We recommend this company: http://www.volantaxi.eu/index_eng.html
If you want to plan your travel to Budapest for a specific location and/or time, you can safely use the Get directions feature of Google maps to plan your travel. Don't forget to choose your preferred way of travel, for example public transport.