You can fly to Kosice airport which is about 50km from Presov. There is bus going from airport to Presov bus station. It's cheap and very fast(about 30min.). To Kosice fly some airlines like CSA, Slovakair, but the cheapest is Skyeurope.
To get to Presov by train is the best. Take InterCity train from Bratislava, you have to change to local train in Kysak. The whole journey takes about 5hours 15min. Disadvantage is that you have to change the train. But I don't recommend you to take a bus from Bratislava cause it takes too long(8-9 hours). There are many connections from and to Kosice(either by bus or by train).
It's the easiest way. But maybe not the fastest because of lack of highways. You can rent a car in Bratislava and follow direction Zilina, Poprad and Presov. From Bratislava to Presov about 450km and 5 hours of driving
Transport in Presov is very easy. Buses go almost everywhere. To get to center of the city from railway or bus station takes just 20min. by walking and 5 min. by bus. 10min. ticket for bus costs 10SKK and 30min. ticket costs 14SKK. To get from one side to the other side of city takes about 30min. by bus.
i think the best way to get to presov from bratislava is by train. you need to change trains in Kysak, which is basically a station to wait at... no big deal. however, slovak trains can be disappointing for most of you.
you can use the townl buses when in town, using the ticket machines at the bus stops. the centre however is small, you can take a look around if you are willing to take a half an hour walk.
Take a freight train from Uzhgorod, Poprad or Bratislava you will save money on accomodation.
Tatra or Skoda