The difference between the black market and official rate is very small, and not worth the risk. The best way to travel to and from the airport is via the Airport Minibus. Use it to get to and from the airport when you arrive. Give me a passport photo of yourself by December 1 and I will get you a transit pass and a Budapest map. Always carry your transit pass - it offers unlimited travel on the subway, busses, and streetcars within the city.
Taxis are safe, clean, and efficient - but the drivers enjoy overcharging tourists.
If you use a taxi, look for the smaller cars with signs such as Fo Taxi and City Taxi. If you are not a U.S. citizen, you may need a visa for Hungary and for any countries you will travel through, such as Slovakia on the way to the Czech Republic or Poland. U.S. citizens do not need visas. Hungary is a safe place with a high standard of hygiene. You don't need any shots.