Hi Julie,
With such good public transportation, everything's within reach. The other major cities around are Budapest, Prague, and Vienna and all can be reached by train or bus.
In Bratislava, looking and wandering around the old city (the most touristy part) only takes a day or so, if you do it slowly and take time to go to museums and stuff. The castle museum is honestly not worth wasting time on. However, there are some good restaurants where you can try some good refreshments and Slovak food.
Near Bratislava is another castle called Devin. All that's left are ruins, but you can go visit it and after visit the winery under the castle that makes and serves a special type of wine made with 'ribezlak' or currants.