Bill has mentioned two that are definitely worth doing - Parc Guell and Sagrada Familia.
With the Sagrada Familia, it's definitely worth paying to go inside - the stained glass windows and the light inside is absolutely stunning. However, we also paid extra to go up one of the towers, and being honest, I didn't think this was worth it.
Another area that is worth visiting is Montjuic. If you go to the Place d'Espanya, you can walk/take the escalators up to the museum, which is a beautiful walk in itself (we've never been inside the museum), and 5-minutes round the back is the Olympic Stadium which is free to walk into. Then further on up are the cable cars and Montjuic Castle - this is a really good way to spend half a day - the views from the top of the castle over Barcelona are superb.
Also, the old town is good for wandering - plenty of old-style labyrinth-type streets.
Have a great weekend - Barcelona is one of the best cities I've visited.