There are quite a few “touristy” spots in Bratislava, though it’s probably not worth spending more than three days in the city. The Bratislava Castle might look plain and uninteresting from the outside, but has a pretty interesting museum inside. I’d also recommend getting lost of in the narrow streets of the Old City, with nice cafes and antique book shops, especially if you’re there in the spring or in the summer. You should also see St. Martin’s Dome and Primatial Palace, all in the Old City.
If you like views, you should go up to the Café above the Novy Most (New Bridge). There’s a small fee to take the elevator, but it’s worth it and it’s not far from the Old Town.
It might also be interesting to take a boat to Devin, an old ruin from 9th century above the confluence of the Moravia River and Danube.
If you’re staying for longer, take a day trip to Vienna, it’s a gorgeous city, and only about an hour away. Trains go every hour or so and are fairly cheap (less than 8 euro return).

1. 1. Bratislava tourist informationcopy