Though it looks old and stately, the Palau Nacional was actually designed for a 1929 exposition. It's part of a reaction against Gaudí and his friends, and its design is not very original. The Palau Nacional houses an art museum and an ethnography museum, which delves among other things, into Roman Barcelona. In front is the Mies van der Rohe pavillion, which is probably one of the most famous buildings of the 20th century. It was the Bauhaus contribution to the 1929 exposition, and one look changed everyone's thinking. Internal and external space are tied together almost without a difference it's simple, yet strikingly intelligent.
Another type of taste is the ridiculous Poble Espanyol a sort of Disneyland version of Spain, built as a tourist attraction for the Olympics. It's replete with dolls, keychains and commemorative plates. The Fundació Joan Miró is just down the road, so keep walking till you're safe inside.
Finally, the Palau is up on a hill, and the views are spectacular.