Barcelona is one of Europe's most visually exciting cities, largely because of one man: Antoni Gaudí. The Sagrada Familia is an active construction site and spiritual masterpiece simultaneously — book timed tickets and tower access well in advance. Park Güell's ticketed Monumental Zone (book ahead) showcases his mosaic architecture; the surrounding free park has city views.
Casa Batlló and Casa Milà (La Pedrera) on the Passeig de Gràcia are two more Gaudí buildings you can enter. The Gothic Quarter's Roman ruins and medieval streets reward aimless wandering. Barceloneta beach is 15 minutes from the Gothic Quarter. Camp Nou (FC Barcelona's stadium) and the Picasso Museum complete a full city experience.