from the airport

if you arrive to budapest by plane, you can easily get to the center, the terminal 1 is quite close to the city, but the terminal 2 is still ok, they are not as far as hahn from frankfurt or luton from london, haha. it takes about 10-20 minutes by bus to reach the blue (3) metro line (map), from where you can get anywhere - for example to go to colors hostel mentioned above, get off at "ferenciek tere". you just have to take the bus #200 from both airports. unfortunately the public transport is very expensive in budapest compared with other capitals in the region, but it's still cheaper than by taxi. a single ticket costs 230 forints, a bit less than 1 eur. but you can't change with it, you have to use a new ticket every times you change.
the controllers aren't very common on this bus. you can afraid of them especially on the metro, so it's better to have a ticket there. (so the full fare is 460, but if you blitz on the bus, just 230 forints)
if you prefer taxi, choose the official of the airport: http://zonataxi.eu. they have fix prices from 14 euros (depend on how far you go) for one way.
an extra tip: you can have your free map of budapest already in the airport, just take it from the shelters until your baggage arrives.