Drink lots of Sangria.
Visit the City History Museum, and be utterly amazed at how they basically ignore hundreds years of Moorish occupation.
Walk around Barrio Gotico (but maybe you don't hate cars as much as me, so this won't be as cool).
Visit Parc de Montjuic, it has nice views of lots of the city.
See Gaudi architecture, wonder if he is a brilliant architect, or just bad Unreal Tournament level designer who is overly in love with curved surfaces.
Eat a tortilla espanola.