If you have a free day, Bratislava has many nice sights nearby, that you can visit by one hour and half or less! Remember that trains are for free if you study in Slovakia, but if you go with other visitors from other countries, don't worry, trains for short distance are still cheap.
You have 3 countries avaliable to visit: Czech Rep, Hungary and Austria, and do not forget about the beautiful Slovak countryside.
-Trencin: Just one hour and half from Bratislava there is the second biggest castle in Slovakia, Trencin castle, located just above the town with the same name, which, by the way, it's quite beautiful.
From the train station, there is just 5 minutes walk to the very centre of the town, it's easy. Once in the main square you have many things to see, like the main church, the synagoge or the executioner tower.
To go uphill, you can just go through any street that goes up, but if you go through executioner tower and walk outside the walls, you will see a nice park at the same time you go up, and when you get into the inner town, you will be in a cozy church with nice views. The castle is around 3 euro for the minitour or 5 for the big one, but with student card you will have good discount. The tour takes from 30 to 90 minutes and is the only option to get inside the castle, so if you just want to make good pictures, the minitour option will be the best.
But sightseeing is not all, if you are hungry but want to save some money you should take a pizza slice in the place which is in the main square, but, if you want to really enjoy your stay in Trencin, you need to go to Lanius brewery, where you can try up to 10 different kinds of local beer, and eat a wide selection of food which includes beer in the recipe.
-Trnava: Actually Trnava doesn't worth a day trip, since is pretty much the same you will find in any Slovak town, but its good location towards Bratislava makes it a nice stop in your way back from further places (like Tatras or Trencin), the main square is quite nice, but is a residential town so don't expect a lot of life there.
-Beckov Castle: If you have ever gone to Tatras or Poland, for sure this impressive castle got your attention, actually is quite easy to reach if you plan your trip carefuly. There is no direct train to Beckov but to Nove Mesto nad Vahom, a city located 7 Km from it, there is a cheap bus (less than 1 €) that connects both places, but check the timetables at www.cp.atlas.sk for the exact times, both bus stations are so close to the destinations, beign the Nove Mesto just in front of the Train Station and a little side road bus stop very close to the castle (about 150 metres).
The castle costs 3.5€ and while it may look expensive compared to other bigger Slovak castles, just think that Edimburg or Stirling castles in Scotland are about 24€... so don't worry about the money. You can actually see the castle in 30 minutes, but there are some nice activities for tourist around (people in costumes, medieval fights), specially in summer (check the web for special ocasions).
The best pictures from the castle are taken from a little high ground just left of the main entrance road (close to the Jewish cementery, it is where the photo is taken). Probably after the castle, you will have some time with nothing to do in Nove Mesto nad Vahom due to the bus/train times, don't worry, it has enought things to keep you amused for one hour, you can see the town centre which is just 5 minutes from the train station (just pass some Soviet blocks and you will see the church tower), there is also a monastery, Eating here is a good option since is definitely not a turistical city (although is well kept) and you can choose almost any place without worrying for the price, in fact you can have Ice Cream for 70 cents in the main square. But if you are too lazy to go to the centre, in the very front of the train station there is a bar where half a litre of beer is less than one euro, perfect to let the time go faster.
-Brno: The second biggest city in Czech Rep. and reaching it is pretty easy, you can either get a bus with Student Agency for around 14€ both ways or you can go by train, which is cheaper (10.8 €). The train station is almost in the city centre, so you wont get lost.
This city has a special vibe, diferent than Prague, but let's start with the city centre: It's bigger than the Bratislava one and is easily walkable in 1 or 2 hours, don't forget to try to guess the time in the weird shaped clock in the middle of the main square (actually, it' pretty hard even if there are instructions). There are other hidden secrets in this city like the two angels having sex statue in the window of the church after the main square or the Brno dragon (which is actually a crocodile).
Once finished in the downpart, you should go to the cathedral hill and see the stunning views from there, but the city is not over... there is a fortress! You must go to the fortress hill and from there you can have the best pictures of the cathedral, also, the fortress is quite interesting to visit, specially the old prison part.
If you want to have fun, go and get a beer in Vytopna where your beer will come by train, literally. But what makes this city so cool is the big ammount of cafes there is, with a lot of styles, so find the one which suits you better and enjoy tea or hot chocolate there.
-Esztergom: In my opinion, it's the best morning plan from Bratislava, you need to get the train to Sturovo, the last stop of Slovak rails (you need to pay 1€ for each way), from there take the public bus (0.20€) to Sturovo itself, since the train stop is located 3km from the town. Once in the town, just go to the only bridge that crosses the Danube and be ready for stunning views. By the way, pay attention because the border between Slovakia and Hungary is in the very middle of the bridge (Check for SK/M sign).
Once you are finally in Esztergom, you can start to go uphill to see the biggest cathedral in Hungary. To do it, check for a very narrow street with stairs (really, is just 1 metre wide). Uphill there are three things to see, the monument in the eastern corner, the Esztergom castle and the Cathedral (it's free!). After you finish with the upper part, you can spend some time in the lower streets and have something to eat, keep paying attention to the time since it can take you 40 minutes to reach the train station.
-Gyor: Going by bus to this nice Hungarian town can be a little bit pricy, since StudentAgency charges almost the same for going here than Budapest (even if its just a 40 minutes trip), so the best way to visit it is by car. Actually you can park well in the centre of the town and see the nice city centre, with several cozy streets and some nice Austro-Hungarian style houses, it's very clean and walking here is quite relaxing, it shouldn't take you more than 2 hours, but is among the best towns you can visit in Hungary excluding Budapest.
I hope you find time to visit all of this places, they are the perfect option for a short not organized trip!
-Viena: The Austrian capital is just 50 minutes away from Bratislava, you can get here by train or by the buses of Slovak lines. You can do this trip for around 12€ both ways. Further information about Viena can be found all over the internet, since is a very visited city, my opinion is that Palaces doesn't worth the ticket (more than 20€) to go inside, since you can see the nice gardens for free, also you can go to the famous Opera for less than 10€ if you book a far away seat (hey, it's something). Also don't be fooled by touristical prices, if you are student your wallet will suffer, avoid pretzels and drinks close to main sights (they can be up to 4€) and enjoy walking the city since... It's free :D. Extra tip: Even if it looks so easy to cheat in the metro, and it cost 2€ each way, DON'T DO IT, if they find you there is no explanation possible and you will pay more than 100€ (they will charge it to your bank account if you try to avoid them).
In this section you will find all you need to become a little expert in the guide place, without expending a lot!