Barcelona is a great place to visit; it has been the object of lots of changes some twenty years ago when it hosted the Olympic Games and it has resulted in a general improvement of the area.
As you probably know, Barcelona is the town where the Catalan architect Antoni Gaudí worked and it can show some of the best examples of his visionary form of art. You can't miss visiting the Sagrada Familia, the expiatory temple which was recently opened for rites but which is not yet completed, or Parque Güell, now a public park on a hill which he meant to become an example of integration between normal city life and art and nature. Closer to the town centre you have Casa Milá, better known as La Pedrera, another incredibly interesting building which demonstrates Gaudí's genius. All these places can be visited (the only free one is Parque Güell), whereas Casa Batlló must be seen from the outside unless you have arranged a special visit or event.
One more thing I would recommend to get the feeling of the place is a visit to the Boqueria, an incredible covered food market where you can buy almost anything and where you can easily get a savoury meal for little money. It is just on the Ramblas, a long tree-lined avenue which goes down from Plaza Catalunya to the port and which is a must-see for all tourists.
One museum which in my opinion you can do without is the Picasso Museum, particularly if you have already seen some of his paintings in other places.