Budapest has a relatively inexpensive public transport system owned by the state with 4 underground lines (’metró’ in Hungarian), 5 suburban train lines (’HÉV’ in Hungarian), trams, buses and trolleybuses. The fastest way to get to a place is by vehicles that run on rails (metro, suburban trains and trams).
Tickets or passes are checked by inspectors who move freely on buses and trams in teams and ask for your ticket. Nowadays in most metro stations, uniformed inspectors stand either at the entrance or the exit and everybody has to show their tickets or passes. With your student card, you are entitled to buy a student pass (valid for 30 days).
You can find the timetable of all means of public transport on the website of the Centre for Budapest Transport (BKK).