One of the best things about Hungary for those who are looking to live on a budget is the public transportation system. Within the towns, there are bus lines that can get you anywhere you want to go. They are reasonably priced and run very often.
Fare dodging is possible, but certainly not advisable. Plain clothes ticket inspectors patrol the public transportation system all the time and if you are caught without a valid ticket you have to pay a fine of several thousand Forints, so it is not worth the risk. If you are using a temporary student card (brown plastic card with a green personal paper card) please make sure to renew it every two months at the Student Services or Erasmus/CEEPUS Office. For this you will need an ID photo and 100 HUF. If you have a permanent, blue plastic student card, you will need to get a new hologram sticker on the back of it at the beginning of each semester. The sticker from the previous semester is valid till the month of March in the next semester.

The main destination for travellers from here is Budapest, of course, which is the nation's capital. Budapest is about 30-45 minutes by train (MÁV), bus (Volánbusz) or special suburban train (HÉV). Any of these will get you into the centre of Budapest and back out again quickly and easily, and for the most part, on time. They each run about once every half-hour, so there is rarely a period of twenty minutes when something is not going to Budapest. The schedules for these are in each dormitory, and also on the website of the university.
Taxis are also good, but they tend to be expensive, and students use them only when they have to. Telephone numbers for the stations, and people to help you arrange these things, are everywhere in the university.