Public transport is very cheap, with tickets from 35 cents for students and 0,7€ for adults, ticket is valid for both bus and tram (try not to cheat since tram officers are not rare and fine is 50€). For checking the routes the best option is to download the App “iTransit BA” (in both iOs and Android), and use the “planner” to find the one that suits you better.
As you settle in, you should get your ISIC card in your University and once you get it you can activate several options, including the transport card. To get the transport card, which will enable you to use all public transport for just 31€ during 90 days you need to go to the transport office, located in Podchod Hodzovo Namestie, an underground passage between Postova street and the Presidential Palace square (Hodzovo Namestie). If a tram worker ask for your ticket, he/she just will check your ISIC card (you don't need to check anywhere in the bus or tram once you have the Transport option activated).
This is the definitive guide for a Erasmus or a Student in Bratislava, all info I know about the city will appear here