It's a city very easy to reach from many places in Europe, as I stated in the Bratislava Cheap guide:
There are many transport options from almost everywhere,, it features it's own airport which is also an important Ryanair hub (17 destinations: Palma, Madrid, Edimburg, Dublin, Girona, Alghero, Trapani, Liverpool, London Luton & Stansted, Málaga, Bergamo, Brussels, Rome, Paris, Birmingham and Athens), is also just 40 min from Vienna airport (it usually works as the second airport of Bratislava, it has a bus to both cities for 7€)
Is reachable by bus or train for a very good price from Prague (10.8 €) and Budapest (8€) also by Warsaw (from 8€ but bus takes 11 hours).
Most buses will leave you in bus station (Autobusova Stianica), where you can get a taxi, bus or walk a little, and by 5 minutes you will be in the Old Town, similar situation can be applied to the Train Station (Hlavna Stianica). If you come by Blaguss bus from Viena airport, it will leave you in Most SNP, the main bridge of Bratislava which points the begining of the Old Town. Finally if you come by plane directly to the city, there is a direct line Airport (Letisko)-Train Station, the bus number 61, but before going to the train station is better to get off in the stop called Trnavske Myto and from there take the tram that better suits you.
Bear in mind that taking a taxi won't be cheap and expect to pay around 25 € if you come from airport and 10€ inside the city, if you speak Slovak, pretend to be a local since taxis are the most obvious tourist rip-off here and will try to cheat you if you are not aware, use them as emergency resource.
Finally you can come also by tourist boat from Vienna from 25 € (not recommended for first time).
This is the definitive guide for a Erasmus or a Student in Bratislava, all info I know about the city will appear here