We absolutely LOVED Barcelona when we stayed there for 4 days before a Disney Med cruise in 2011. I think the smartest thing we did was to spend an entire day on the HOHO bus, getting to know how the city was laid out, seeing some sights from the bus, and getting off of it to see specific sights of interest, like Park Guell.
If I were to list our 3 must-sees, I'd say:
La Sagrada Familia (go inside. It is worth the wait. Do the audio tour -- there is so much you would never notice and not know if you did not listen to the audiotour. Be sure to go down into the basement to see lots of the original plan drawings, construction photos, scale models, etc.) Walk around the outside and truly appreciate the different facades and the ever-evolving architectural and artistic styles that have resulted from this being an ongoing project that has been a work-in-progress for generations.
Park Guell - What a fascinating, beautiful park. The views of the city from the upper levels is amazing. Read the story of how the park originated and what it was supposed to be. Gaudi was a true visionary and his influence is seen throughout the city (check out the street lamps near Casa Battilo)
Casa Battilo - Wait in line. It's worth it. The outside is picturesque and fascinating. The inside is really amazing. The use of light and the architecture is something you will never forget. Go up to the roof and look out over the city. Notice the shape on top of the building and hear the story of what it represents.
Barcelona had an energy and beauty that is so unique among all of the places we've been. I dare you not to love it! (BTW, beware of pickpockets in crowds and especially while watching the street performers on Las Ramblas.)