First steps

If you are about to travel to Hungary, there is some sites you should check out:
http://www.mav-start.hu/english/index.php for the railway guide, if you are about to discover more from this country.
http://www.menetrendek.hu/cgi-bin/menetrend/html.cgi for the bus timetable, Volan is the major bus company here. Sadly there is no english version so what you should know: Honnan means from and hová means to, and keresztül is the place where you should write what town you want the bus to go through, you can leave it empty if you just want to go from A to B. Dátum means date, its only important if you want to go on weekend or during the week cause there are two timetable for buses, for weekend fewer bus cause less ppl are going to work.
http://www.bkv.hu/english/home/index.html is about public transform in Budapest. Its quiet helpful and good site.
http://www.bkv.hu/english/jegyinfo/fares.html this is the prices of the tickets. If you stay a but longer and travel a lot you should buy things like the 3 day pass, but the Discount coupon book (contains 10 tickets) is good if you wont use many public transport, but be sure to have a ticket every travel cause the penalty is high and the controllers doesnt really speak any foreign language. (later i'll post a language "lesson" with the basic words. I suggest to have a notebook and a pencil always in your pucket, so you can write down the words, or for example if you want to buy something the salesman can write you down the price)
http://utvonal.bkv.hu/ is for planning your direction and i mean when im saying it will be your best friend if you wanna travel around Budapest... 

story

So MÁV is the train company in Hungary, it can take you to everywhere. Volán is the same for bus. BKV is the company for Budapest public transform. There are some yellow buses, or white they are volán buses, you cant use the BKV tickets on them. With BKV you can use the tickets which prices are above in the link i gave you. It doesnt matter if you use for one stop or for 10, one vehicle use is one ticket, if you change vehicle/train/subway you have to use another ticket, but when you have a pass you dont have to worry about that. BKV also takes you outside Budapest, to the nearest towns. It works this way: while you are inside Budapest you use a ticket or your pass, but in HÉV's (little train or suburban) there are the stops over the doors and there is a line showing where the outside area starts. From there you have to buy a ticket from the controller if you meet her/him. Its a bit strange and difficult i know but this is how it works. Later i post about Mogyoród (Forma 1, Aquaréna and its history, cause i live there :P) and about Gödöllő and the Grassalkovich castle. Till that post your question.