Love Barcelona! Just went for the first time. Here are a few favorites:

• Parc Güell
  
• La Pedrera and Casa Batlló
  
• Mercat De La Boquería - the juices are wonderful!
  
• Museu Picasso - nearby Demasié is a great place to get a treat.
  
• La Rambla
  
• Barceloneta
  
• El Born - spend a day walking around here, it's awesome.
  
• Don't forget to try churros + chocolate, fresh orange juice, the sangria and jamon
• Grab a drink on a rooftop bar. W Barcelona is actually a good spot.
  
  

I wrote a bit more about things to do/not to do once you get to Barcelona here (with some specific recommendations). We had some interesting learnings on our trip!

Enjoy :)

Depends on what you want. The touristy sites, like the Sagrada Família, and all of Gaudi's works like Casa Batlo, La Pedrera are great. Definitely tour the Gothic Quarter, Barcelona's most historic area and you can walk on La Rambla, and head to the Raval neighbodhood which is quite cool and rustic and less touristy. The beach area is great as is Montjuïc and other neighborhoods like the El Born (Trendier neighborhood in the Old City) and Gracia, which has a more local vibe. I'd do a mix of neighborhoods and highlights. If you're into Museums the Museu Picasso is worthwhile as well as the 'Ideas Museum' in the Gothic Quarter and then other's like the CCCB and the MACBA, both in the Raval. 

Hello Dave. Barcelona is in one word: stunning!! Gorgeous weather, incredible architecture and the perfect mix of city and beach. Having said so I would detail the best places that you must visit: Starting with Plaça de Catalunya that is at the top of La Rambla the main street that leads to the ocean. Once you see Christopher Columbus statue you have reached Port Vell and Maremagnum. Amazing places to enjoy the sunset. Another Interesting and must see places are: La Pedrera, Casa BatllóSagrada Família, Parc Güell, Museu Nacional d'Art de Catalunya, El Barri Gòtic, Montjuïc, Museu Picasso and Barcelona Cathedral. Barcerlona is a city walk, observe and enjoy. Hopefully you can take the Barcelona City Tour located at Plaça de Catalunya.

Hope this helps!! Saludos

Take a day trip to Montserrat, the monastery in the sky, it's breathtaking and likely not like anything you've seen before. 

Also make sure you see the Sagrada Família and go inside, it's a monumental church and it's so different from the rest -- pre-buy your ticket though!