Budapest:
New in 2013: 5/30 transport pass
This pass entitles you to city transport 5 times 24 hours of your choice within a 30-day validity. Also a day ticket for small groups of 2-5 persons is available at the prrrice of 2 individual day tickets.
Simply buy one of the cheap network tickets for the appropriate time period: 24h, 72h or 7 days, 48-h family pass, small group day pass for up to 5 persons, or a 2 weeks, 5x24h/30 days, or monthly pass (called "bérlet"). Most tickets and passes are valid also for railway and Volán coaches within city limits. This will save you money and hassles given that the single ticket system is pretty complicated. Budapest 24h day passes are also sold by drivers of 200E airport express bus and conductors in the suburban HÉV light rail system (lines H5,H6,H8-9).
Single tickets are valid for one trip only using one mode of transport, even if jumping the tram for two stops. (Except on the Metro where a single ticket is valid for 60 minutes including any changes within the metro system.) So if you have to change several times, with single tickets you will soon end up paying as much as in the world's most expensive cities. If you prefer to walk around at your own pace jumping on the occasional trolley bus, underground or tram as much for the experience as the transport, buy a coupon book with 10 single tickets. There are also short distance metro tickets for up to 3 stops and "transfer tickets" for two rides with a change on any means of transport.
The Budapest Card is a good option if you can take advantage of some of the free or reduced admissions this card affords.

MÁV railway and Volán long-distance bus services are a good option for some relations within Budapest and if travelling to the outskirts. With a Budapest transport pass you will have to purchase a ticket only from city boundaries to your final destination. By train you can reach most suburbs and some inner city destinations in a fraction of the time compared with the city buses and often without changing.

Other main cities:
The transport companies of major cities such as Debrecen, Miskolc, Szeged and Pécs operate trams, buses and trolley buses. Network passes can be bought at reasonable prices. In other towns the public transport is run by either a local bus companies or the Volán Bus Co. which also operates the coaches between towns. Interestingly until the recent past, narrow gauge railways played a part in urban, local and recreational transport in places such as Nyíregyháza and Kecskemét (unfortunately closed down in 2009). Narrow gauge railways continue to operate in cities like Debrecen, Miskolc, Gyöngyös and Pécs with a recreational/touristic focus.
Hungary Card affords two free day passes for several cities, next to a half-price Budapest Card and many more benefits.

Go to timetables (including route planners and city transport system websites)