The Palau de la Musica is an amazing building. Guided tours are offered on a daily basis, but the best option is to try and get tickets to a concert during your stay in Barcelona. The Palau is easy to find, just off Via Laietana, (nearest Metro: Urquiniona) and a short walk from the Cathedral.

Hospital Sant Pau is an excellent example of modernist architecture. It is free to enter the gardens, where you can observe the different buildings which are still in use. The nearest metro is on the blue line (Sant Pau), but I would recommend you combine this visit with one to the Gaudi cathedral (La Sagrada Familia). The two sites are close. You can walk from one to the other along Avenida Gaudi in less than 10 minutes.