La Rambla is the first place most visitors head to in Barcelona. This tree-lined pedestrian mall in downtown is the hub of the city's activities. It starts at Catalonia Plaza (Placa Catalunya) and ends a mile later at the sea. Along the avenue is a hodge-podge of street vendors selling flowers, birds, coffee and everything in between. Don't miss La Boqueria, the amazing market. It's one of the best in Europe.
After stretching your legs along La Rambla, it's time to see a little architecture. Anton Gaudi was a famous early 20th century architect who was responsible for many of the interesting, and very different, buildings in Barcelona. His most famous work, the La Sagrada Familia cathedral, is still under construction. Other Gaudi works in the city were completed and open to tourists, including Parc Guell and La Pedrera. I especially enjoyed touring the Batllo home, which is now a museum. I wouldn't want to live there, but it provides an excellent overview of Gaudi's work.
If you are in Barcelona on a rainy day, the city has several excellent art museums, such as those featuring the works of Picasso (who moved to Barcelona when he was 14) and Joan Miró. Football (soccer) lovers would enjoy the FC Barcelona football club museum, which is the fourth most visited of the museums in Barcelona.
I've just provided a few options of things to do and see in Barcelona. One other note -- the city really starts to come alive after dark. People dine very late in Barcelona, so you'll have plenty of time to tour during the day and dine and party all night!
La Sagrada Familia in Barcelona (c) Linda Garrison