Sagrada Familia: this church in progress is unlike anything you've seen before. It is truly spectacular. Buy tickets online beforehand.
Hospital de Sant Pau: one of my favorite places in the city and very much under the radar of the hordes of tourists. I discovered it because I lived right by it but there is actually a pedestrian walking street that goes from Sagrada to Hospital. Definitely do free the tour, which explains the history behind the buildings and gives you a bit of insight into Lluís Domènech i Montaner, one of the most important modernist artists, right up there with Antoni Gaudi.
Palau de la Musica Catalana: another one of Montaner's buildings that is absolutely stunning. There is a tour or you can buy tickets to one of the shows.
Bario Gotic: this is a fun neighborhood to explore. It has great boutiques and small restaurants.
L'Eixample: Passig de Gracia is another good shopping street in the L'Eixample neighborhood. But wander off the main street to see some beautiful buildings, cute little bars and shops.
El Raval: this is a mostly student and immigrant heavy area but it was once the most important parts of the city so you will find the Boqueria Market and several other Gaudi designed houses here. If you are lucky, you will run across El Gato de Raval, an awesome life size cat sculpture from Colombian artist Fernando Botero.
Firo Tast: this tiny little tapas bar is located on the street that runs between Sagrada Familia and Hospital. We ordered a wide range of items and everything was delish!
Spoonik: make a reservation in advance for this tasting menu only joint where the chefs personally describe the different ingredients and inspirations and the lighting, music and art changes depending on the course. You also sit with strangers so hopefully you get a good crew like we did!