It's a great mistake to try to drive around Budapest1 in a car. It's a big city with all the disadvantages that can bring, but local public transport is very good. Daily tickets are cheap and a range of other ticket deals are available to help cut costs.
If you arrive by plane, the best way to get to nearly anywhere is by the Microbus2 system from the airport. This form of transport will take you to anywhere in Budapest for about 600Ft3 which is less than a third the cost of taking a cab. Alternatively, the 198 bus will take you to a number of underground stations to connect with the system's blue line.
There is a system of underground or Metro lines from each train station. Don't try to save money and sneak on to the underground without paying - the conductors are very mean, they don't speak foreign languages, and they can always be found around Deák tér station. Unlike any other city, Budapest underground tickets are not valid for transfer and Deák Ferenc square is the only place where you can transfer Metro lines.
To get around the countryside, most people travel by train, which is slow, but not as slow as travelling by coach. If you don't know what your station looks like, ask someone to warn you when you are there because there's no announcement or any signs when you pull into a station.
In most cities - Budapest is the exception - buses only open the front door after 6pm or 7pm, so don't be taken unawares and head for the centre doors.
Finally, when travelling around the country, be wary of some of the sights that are no longer 'Hungarian'. Lake Balaton is now a resort for German tourists - most signs are written only in German and the restaurants don't serve authentic Hungarian food. Furthermore, it doesn't even look like Hungary.