Budapest airport has a good page on its website
Public transport to Pest is pretty easy, but it appears getting to Bud can be tricky with multiple metro and bus trips.
There are shuttle bus/mini van services which are about $15 per person (or even less).
Budapest taxis (Fo taxi) are reliable and have a fixed rate for the city - about $25.
Check out our other travel tips for Hungary here: http://safety.worldnomads.com/hungary
Phil at the safety hub. about 6 years ago