Mercat de Sant Josep de la Boqueria. I bet food here is more expensive then somewhere else but if you're not gonna bye weekly supply of food, if you just wanna eat fresh paella or fruit salad or take sweets for relatives - welcome.
La Sagrada Família. Amazing light. Churches and cathedrals are one of my biggest passion (???). This one is absolutely another way of understanging the word 'cathedral'. The way to understand religion as weightless, light, ethereal mediator between Person and God. Even if you're unbelieving you'll feel it, I guarantee.
Museu Nacional d'Art de Catalunya in Palau Nacional. It is probably one of the best well-organized museums I've ever seen (in fact I've seen not so much world-famous museums, especially well-organized). It's very comfortable and friendly for visitors/ For example, your ticket is a two-day visit during the month. I wouldn't say that collection is very special (it's not Prado or Hermitage) but anyway significant as it's the biggest collection of Catalan art in the world.
The best day in Barcelona was not in Barcelona. We tried to bath in the sea on Barcelonetta beach and even in outskirt. But in the first case it was too crowdy and dirt water, the second case it was too dangerous (perhaps) cause of lot of strangers there. So the best decision was to go to little town one hour from the city, St. Pol de Mar. Adorable.