I'd recommend going to Amsterdam, it's unique and unlike anything you've ever seen. Of course I'm from Amsterdam, but that left aside, it's not for nothing that there's SOOO many people that come visit the city ever year.
If you end up coming here you can shoot shoot me a PM - you can crash at my place if you want. Rooms here are crazy expensive, 50€ a night if you're lucky, which is about 70 bucks or so.
The thing you need to do is go INTERRAILING. It's basically a train ticket that let's you travel anywhere throughout Europe, whenever you want, wherever you want. It's the ideal thing for backpacking - you'll be able to see pretty much whole Europe for relatively cheap! If you're gonna end up buying a ton of train tickets it's gonna cost you a fortune.
Towns definitely worth going to:
- Amsterdam
- Paris
- Berlin
- Barcelona (you're already going there, yeah!)
- Venice
- Rome
- Prague
All VERY beautiful cities. That's more for the sightseeing I guess, if you want to settle somewhere for a longer time, I could recommend the southern parts of France for sure - the most rustic, serene place on earth probably. It's idyllic and super nice.
Like someone else already said, go to Croatia - I've never been there, but it's everything you could want from what I've heard:
It's cheap, incredibly pretty and a very nice climate.