Any trip to Barcelona would not be complete without some stops at very touristy places. And if you're in Barcelona, tourism means: Gaudí. He is everywhere. The architect/madman/visonary of La Sagrada Familia had a hand in many Barcelona pies. He is unconventional, inspired by nature, and everything in his wacky, gorgeous designs serve a purpose. The tickets to visit the different Gaudí buildings are pretty steep though, about $20 a pop, but in my opinion, totally worth it. You can't really understand Barcelona, methinks, if you don't have an appreciation and somewhat understanding of Gaudí.
La Pedrera/Casa Mila
La Sagrada Familia