For one, you must definitely go and see La Sagrada Familia cathedral designed by Antonio Gaudí. You'll get filled with respect and admiration for this man whom you never knew but who conceived, almost a century ago, a project as surrealistic as this.
Although the cathedral is still under construction (financed by donors just like you!) and I believe, will remain so for another 50 years (at least), you can already see the outline of the magnificent four-sided façade. It has so many details that you will probably end up taking at least 1,000 pictures if not spending the entire day looking at it and putting all the pieces together in your mind, like some sort of gigantic puzzle, trying to grasp the magnitude of the edifice.
But this is not the only building Gaudí bequeathed Barcelona by. Casa Batlló, which is my favorite building in the city, looks like it was inspired by the form of a dragon. It is several stories of pure Art Nouveau and once you're in (after 18 Euros and possibly a line), you'll probably feel the way Alice must have felt upon her arrival in Wonderland.
The construction doesn't have corners - everything flows and undulates gently, letting you admire, in silence, the cute fireplace, the ventilation system that looks like gills, the chandelier that must have been owned by a sorcerer and the most beautifully designed laundry room you'll ever see. The only annoying thing is other visitors who get so immersed in listening to their phone-guides telling them how beautiful the building is that they forget that it's better to simply look around leaving history readings for later, rather than stand there obstructing everybody's view. Those evil earpieces seem to control them by whispering in their ears: "Step into the space he's trying to photograph - yes! good! now, stay there!"
Gaudí's house is also in the park, but I wouldn't particularly recommend it as there's not much to see inside besides a bunch of Gaudí-built furniture collected from various other houses he designed (Gaudí never used them, he only created them) and a few pictures of the great master. The only remotely Gaudí-an thing is his single bed in a sparse room, for once telling you something about his life rather than about his work (or are these two things one and the same?).
A map of Gaudí's works in Barcelona: