As previously mentioned you really can walk to majority of locations that the average visitor would want to go in Budapest in under 45 minutes. Another option is a multiple day transit pass:
http://www.bkv.hu/english/jegyinfo/fares.html You can ride for 3 days for the taxi prices mentioned. I have only used Zona Taxi to / from the airport but they were good.
http://www.zonataxi.eu/en/