The first thing you will experience is probably the airport. Budapest has two of them, Ferihegy 1 and 2 (terminal 1 is closed at the moment). They are both close to the city, and you can get from there to the city with Fő Taxi (no set prices anymore), Airport Minibus (3200 Ft) or you can travel with public transportation, but if you come for the first time that is not recommended. The taxies will be waiting for you outside the airport, and the Airport Minibus can be requested as well upon arrival. The difference is that if you choose the minibus you will travel with others, and it will take some more time.
If you would like to use public transportation you can travel with bus 200E from both of the airports. Be sure to jump on in the direction of Köbanya Kispest, which is the endstation of your bus trip. You can buy a singleticket at the bus, or you can buy at the Relay (papershop) inside the airport (recommended). When arriving to the endstation you at the same time arrive to the endstation of the blue metro line. From here you travel with the metro (you can buy tickets/passes as well) to where you want to. Check out the metro map, and what stop is closest to your hotel or apartment.