The city has its own bureau of tourism in which official website you can find all the information you need BarcelonaTurisme.
The Ciutat Vella and Barri Gòtic districts concentrates most of the historic interest like the Picasso museum, the Modern Art Museum of Catalonia, the the National History Museum, or the Center for Contemporary Culture of Barcelona, among many others. The Avinguda Maria Cristina in Montjuïc also holds the interesting National Art Museum and Cosmocaixa.
Antoni Gaudí's major architectural jewels are spread around different districts: from Passeig de Gràcia with La Perdrera and Casa Batlló to Gràcia with the beautiful gardens of Parc Güell and L’Eixample where Sagrada Familia stands. To enjoy the sea front La Barceloneta, the old times fishermen district is now opened to the sea through a vibrant and modern promenade where one can stroll, skate or ride a bike.
Barcelona and its citizens have always been very keen on sports. The Olympic hive the city experienced in 1992 is still present. Athletics, Basketball, Tennis, Sailing, … are some of the major sport events besides football. If you are a football fan you will enjoy a visit to the Camp Nou.
If you intend to stay longer and visit other parts of Catalonia, the Tourist Bureau of the Catalan Government covers in detail the different activities and places of interest. It also provides links to the specific tourism bureau of the different locations.