parc de guell is a must-see. It is a park designed by Antoni Gaudi - who also designed the Sagrada Familia (the big/important religious temple).
Also, I very much enjoyed going to Parc de la Ciutadella. There's a fountain that you can sit up at the top of, and on a good day (which it will be at the end of April / early May) the snack vendor is out and he sells beers.
During an afternoon with good weather, go down Las Ramblas to see the street performaers. There's also a tapas bar at the beginning of Las Ramblas (it's on the right just as you start on Las Ramblas) which is definitely worth a stop. There was also a wonderful flower shop that had many different colors of roses, which even for us guys is kinda neat to see such colorful variety rarely (in my case never) found in the US.
If you're into museums, there's an average Sex Museum (not as good as the one in Amsterdam), and I did enjoy half of the Picasso museum. It's free on the 1st Sunday of the month and closed on Mondays.
You can skip the aquarium unless you've got nothing better to do. They had a penguin. That was nice. Other than that, it's rather small.
When I was there, I stayed with a local which really enhanced the experience. Whenever I travel alone for pleasure, I always find locals on craigslist or just by searching, and every time it's been a wonderful experience. There's nothing like having an instant inside scoop.
You can walk into a McDonald's just to witness the menu... particularly of note is that they sell beer there.
Public transport is good there, and the Information Center is very helpful (free maps, too!).
Other than that, the most enjoyable thing about Barcelona was just wandering around.
If you're going to stay for a while in Spain, then take a weekend down in Seville. It's just a delightful little town with a lot of Spanish flair. That's where you'd see authentic Flamenco dancing, as well as Christopher Columbus's tomb.
And if you're debating Madrid, skip it. It's a big city, nothing interesting, and the locals will take tourists for a ride. No, literally. The taxi drivers will take the "long" route even if your location isn't very far they find a way to jack up the fares. And theft is a bit more rampant in Madrid (it' the only city in Europe where somebody attempted to pickpocket me).
FYI -- when people try to sell you things on the street (like watches) particularly on Las Ramblas, don't even give them the time of day. There's a lot of tourist scams specifically for people who see that you're a tourist and stop you specifically to sell you something at a great price. Everything from bait-and-switch (box really has nothing of value), to having kids pickpocket you when you're distracted by the seller.