Budapest:
From 2011 24h + 72h as well as weekly city transport passes are valid for train and Volán bus services within city boundaries. Also it is now prohibited to smoke in bus, tram and rail stops.
Simply buy one of the cheap network tickets for the appropriate time period: 24h, 72h or 7 days, 48-h family pass, or a 2 weeks or monthly pass (called "bérlet" and valid also for railway and Volán coaches within city limits). This is guaranteed to save you money and hassles given that the ticket system remains pretty complicated.
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). 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 are a tourist and are able to take advantage of some of the free or reduced admissions this card system affords.

Season "bérlet" tickets are also valid, within city boundaries, for MÁV railway and Volán long-distance bus services - a good option for some relations within Budapest and if travelling to the outskirts, when 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 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 and network passes can be bought at reasonable prices. In other towns the public transport is run by either the local bus companies or the Volán Bus Co. which also operates the coaches between towns. Interestingly narrow gauge railways presently play a part in urban, local and recreational transport in certain places such as Nyíregyháza, and Kecskemét (unfortunately closure seems a matter of time), Debrecen and Miskolc (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)