From the Airport

Ferihegy International Airport is located 24km southeast of Budapest and has two terminals side by side and an older terminal 5km to the west. To get into the city there are a few ways but the most popular is the Budapest Airport Minibusz (+361 296 8555, or through your travel agency). The minibusz is a nine seat van that ferries passengers from the airport directly to their hotel/hostel. Prices vary according to what area you are residing in; but in general the city centre is priced at 2990Ft one-way, 4990Ft return. Be aware that you need to book your return journey at least 12 hours in advance; try not to leave it too late to catch your plane as they’ll probably be up to eight pick-ups en route.
In terms of taxis, Zona Taxi (+36 1 365 55 55) has the control over this situation. Chosen by the airport for its low fares and ‘highest levels of service’ Zona taxi fares start at 1700Ft and climb to 5800Ft depending on the city zone you will be accommodated in. Of course you can take any taxi to the airport; and some even have a discounted rate.
Since July of 2007, you can travel easily between Ferihegy Terminal 1 and the Western Railway Station in Budapest.  Fifty one trains operate on the weekdays and at weekends 38 trains are in operation. The travel time is less than half an hour at a cost of 300 HUF. Thus, making it the cheapest alternative. You can buy your tickets at Tourinform at Terminal 1 from 9:00am-10pm or at platform B at the railway station from the ticket vending machine.