The public transportation is operated by BKK. All basic information is available on their website. In Budapest tickets and passes are also available. The whole territory of Budapest functions as one zone.
Important to highlight that tickets are only valid for one ride and only in one direction, independently of the stops you are taking. This means that when you change between vehicles or means of public transportation you must validate a new ticket! The only exception is metro, where you can change freely and without any further payment between the lines. Consequently, if you are willing to use public transportation often, it is much easier to purchase one of the passes. Passes are available for 24, 48, 72 hours or one week. Budapest Card, which gives you various discount, is also a good choice. To buy these you will need a valid document which number will be printed on the pass.