MACBA, the museum for contemporary art in Barcelona, not only has impressive exhibitions. The building itself is amazing too. The square in front of the building is popular among skaters. Close to the MACBA, you´ll find CCCB. This centre showcases cultural projects of an international calibre and has a great shop where you buy (art) books and designer gadgets.
The Caixa Forum (close to Plaça Espanya) is also worth a visit and the entrance is free. Not far away from Caixa Forum, is Mercat des Flors, a theatre for performing art.
The Centre D´Art Santa Monica has a lot of temporary exhibitions and from here, you also have a nice view over the Rambla.
Antoni Gaudí is undoubtedly Barcelona's biggest influencer. You'll see the undulating dragon's back of the Casa Batlló, the piercing spires of the Sagrada Familia and the avant-garde mishmash of ideas in the fabulous Parc Güell.
The Foundació Miró has the biggest collection of Joan Miró's paintings and a spellbinding view over the city from Montjuïc.
And last but not least: Museo Picasso. Definitely a must see when you are in Barcelona.