Budapest has an efficient, extensive and inexpensive public transport system and there is absolutely no reason not to use the many trams, buses, trolley buses, trains and metro system on offer.
All public transport requires self-validation of pre-purchased tickets so it is far easier, and cheaper, to get a travel pass on arrival. The Budapest Card, which gives you two or three days of free travel with discounted fees for attractions, is extremely good value for any tourist.
Most transport runs until about 11.30pm, after which there is a limited night bus and tram service until about 5am, and if you have been unsure about travelling on foreign underground systems in the past, rest assured that the metro in Budapest – whose three lines will take you to most areas of interest - is clean, safe and frequent.
Trams are good for travelling around the Great Boulevard or along the embankment and although more difficult for foreigners to get to grips with, buses are useful around Buda. In Pest, take a trolley (or electric) bus.
Note that one of the few ways to reach the Castle district, other than on foot, is via a funicular and if you are heading for the city suburbs, you will need to take an overground HÉV train.
With plentiful buses from the airport, you will probably never need to shell out on a taxi. If you do, avoid the private, unmarked vehicles that hang around all the key stations. Legal taxis have a yellow number plate, clearly display their rates and unlike the rogues, have a meter that is switched on.