Bratislava is basically split into two areas, the old town and the new town. The old town is quite beautiful with market stools, nice restaurants and is quite touristy and the new town is the typical city with shops, eateries and memorials. Both are wonderful. My friend and I literally walked EVERYWHERE, we didn't need to spend any money. There is a lot of history to the place and lots of secrets to discover. I recommend going on one of the walking tours as you learn about old traditions too.
Insider Tip:Try the local Garlic Soup - it comes inside a large bread roll which is fun to eat. Share it if you can, it's incredibly filling!
Walk up to the castle, it's quite plain to look at but the views are incredible.