Barcelona in 2-3 Days

Casa Mila, Parc Guell and the Picasso Museum all require timed tickets that you'll need to buy in advance to avoid standing in extremely long lines and possibly not getting in at all. Parc Guell tickets are not even sold at the park itself. It's possible tickets for the jazz evenings at Casa Batllo also need to be pre-purchased.

I don't recommend the Picasso Museum for folks who aren't serious fans, because it tends to be grotesquely crowded to the point that it's hard to get close enough to the art to read the labels mounted on the wall. The Museu Nacional d'Art de Catalunya (MNAC) on Montjuic is large but doesn't draw such crowds. The Miro Museum, located nearby, is also more pleasant to visit than the Picasso Museum.

I recommend figuring you'll spend a minimum of 90 minutes inside Casa Mila. You may not get inside right at the time your ticket indicates; they seem to meter people in and out. However, lunch is very late in Barcelona, so you won't miss that meal unless you have a significant problem buying a suitably-timed ticket.

Ticket prices for a lot of the sights in Barcelona are quite high, so I recommend thinking carefully about what you plan to do before finalizing your plans and buying the tickets. If you want to leave some time open, you can probably walk right up to the ticket counter at the Sant Pau Modernista site and go right in. It's a multi-building site with attractive grounds. I spent a bit over 2 hours there, but I am unusually fond of modernista architecture.

Are you a fan of Gaudi? I am not, though I love the early modernisme architecture and decor. The only Gaudi sight I've visited (3 visits, total over 2 weeks) is the Sagrada Familia. I've seen some of the others from the outside. I walked past the Picasso Museum one day and there was a very short line for tickets so I decided to go in. It wasn't uncomfortably crowded (it was in March) and I found it moderately interesting, mostly for the development of his art from his teen years. I don't remember there being any major works and significant periods of his work weren't represented at all.

What do you want to see and do on Montjuic? There's a lot there, 2 major art museums, 2 other museums, a fortress (called a castle), the fountain and lovely gardens with good views of the city.

I agree. The tourist sites were my least favorite part of the visit. This may be useful.
The Washington Post's A local’s guide to Barcelona

Wanted to thank everyone here for their advise. We just wrapped up our trip, and absolutely loved each and every one of the cities we visited. Its probably one of my favourite countries to visit now and I plan to return soon.

We tweaked our itinerary a bit from above. Did the two Casa's the same day as Sagrada Familia (8:30 (tour) -11:30), followed by the Park and the two Casa's in the evening one of which was the Magical Nights. Did Montjuic the day we arrived (we got in early at 10 AM) so had a majority of the day to spend there. This gave us one whole day to just wander around in the Gothic Quarter, Las Ramablas, the Gracie neighborhood etc. If I had to change anything then it would be to miss Casa Battlo inside tour and maybe skip Park Guell, spending even more time wandering around. The Washington Post article link provided below to spend more time walking around was probably the best thing I read before I left and it helped me 'plan slightly less, and enjoy more'.