There are three main ways to travel between Budapest’s Liszt Ferenc Airport and wherever you may be staying: public transit, shared airport shuttle, or private cab.
Each mode of transportation is outlined in detail on this webpage: http://welovebudapest.com/budapest.and.hungary/how.to.get.to.downtown.budapest.from.the.airport
One recommended way is to take a miniBUD shared airport shuttle. They provide door-to-door service at reasonable rates. Their website: https://www.minibud.hu/en
Budapest boasts an intricate public transit system composed of above-ground trams, trolleys, buses, and an underground metro system. Each mode of transit requires the same ticket, available in stacks or passes from any BKK vending machine. These ticket ATMS are located at major transit points, or underground, in the BKK transit offices. Using public transit is the cheapest and often the quickest way to move through the city. You can find a list of ticket and pass types with set prices here: http://www.bkk.hu/en/tickets-and-passes/prices/
Finally, Főtaxi is the local taxicab mainstay, with 24-hour service and a smartphone app to order a car. Information at: http://fotaxi.hu/?lang=en#
Any questions may be addressed to:
Michael J. Thompson at ThompsonMi@wpunj.edu, or
János Kelemen at firstname.lastname@example.org