Places my friends and I visited:
1, Segrada de Familia
2, Park de la Ciudadella
3, Arc de Triumf
4, El Carro de l'Aurora
Spent too much time in Segrada de Familia (almost 3 hours), but totally worth the time. Go early to avoid the crowd. Walk up the tower if you can. If you take the elevator, you will miss some of the best view of the city and photo opprotunities. Take the tour if you can, she will tell you the history of the temple.
Park de la Ciudadella is like the Central Park of New York. Lot of locals minus the tourists. Go check it out if you have the opportunity.