This is a tour of some of the major sites of Barcelona and is a complement to the Las Ramblas/Gothic Quarter area. Starting near the Design Museum, pass by the famous Sagrada Familia. Then, the Eixample area with some famous architecture along the Passeig de Gracia, Modern Art Museum, the niversity, Arc de Triomf, and the National Theater. Of course you can modify this and add on streets as you like, such as the Parc de Ciutadella (from Arc de Triomf), and the Gothic Quarter just to the southwest.
Note: Streets can be busy and crowded — go early!