It is very easy to get to the city center from Budapest Airport. Of course the most comfortable is taking a taxi in Budapest, which is Fotaxi at the airport, but there are several other options. The cheapest is by bus, metro and train. Here is a step by step guide how to get from Budapest Airport to the City.
Don't forget that with the closing down of Terminal 1 or T1 for short on May 29-30, 2012, you will need to calculate your trips from Terminal 2 (a short ride from Terminal 1 by car, taxi or bus).
There are all sorts of ways to get to the City from big budget to tiny budget, from standard day trips to late midnight or past midnight rides. You can use the ATM at the airport and you can buy tickets at the airport, so no need to worry about money issues either. Have a nice flight and have fun in Budapest, this wonderful city with immense hidden treasures and cool things to do.