The public transportation system in Budapest is safe, cheap, and extended. The student package contains necessary maps and descriptions. Tickets are valid for all public transport vehicles: subway, buses, trams, electric street cars (“trolley bus”). Transportation maps on the Internet are available from here.
For accessing any city destination from the Airport, the best value for money is the “Airport Minibus”, which operates as a multi-customer taxi, and costs 2500 Forints (15 USD).