I'm currently in Barcelona, and I asked the Tourist Office how early I needed to book admission to La Sagrada Familia and Park Guell, given that I preferred early AM times for both. (Folks have reported that the church is a bit less crowded early in the morning, and heat can be a factor at the park.) I was told one day ahead for the park and two days ahead for the church. Note that the T.O. sells tickets for both, adding a 2-euro convenience fee, but not tickets for the La Sagrada Familia towers. I don't know how far in advance you need to book the towers. Since your time is limited, I recommend getting PG and LSF tickets online before you leave home.
There's a little gatekeeper's house at the park, designed by Gaudi. There's a separate line within the park to go inside that building. If you want to go inside it, try for the earliest park admission time, enter at the main gate, and go immediately to the building on the right. The line was over an hour long when I arrived at 8:50 today, so I skipped that visit.
You need to avoid the long ticket line at the Picasso Museum either by purchasing your ticket ahead of time or by taking a tour that includes the museum. The T.O. runs a Picasso Tour Tuesday through Saturday at 3 PM. You see a few spots in the old city with Picasso connections then go to the museum (entrance fee included), where the guide provides background info on some of the paintings. The tour is $22 for adults. It runs about 2 hours, and you can stay in the museum after the tour if you want to. It's not a tour for people looking for detailed info on Picasso's art, but I enjoyed it, and I think it would be especially worthwhile for folks not planning to take a general tour of the old part of the city. Warning: The museum is very popular and thus very crowded. It's free on Sunday from 3 PM to 7 PM, but the conventional wisdom is that it's even more of a zoo then.
The Museum Nacional d'Art de Catalunya is perhaps the top sight on Montjuic. It is free on Saturdays from 3 PM to 8 PM. The T. O. told me that going on a free day would probably be OK, because the museum is large and people spread out. Can't tell you from personal experience, though. The museum is closed on Mondays.
The Fundacio Miro, also on Montjuic, is a bit pricey (12 euros, plus 5 euros extra if you want the audio guide), but certainly worth it if you're a Miro fan. Otherwise, probably best to use the time elsewhere. (I'd suggest the even more expensive Gaudi site La Pedrera; Casa Batllo is also very interesting but more crowded. Or if you are tired of crowds, another really lovely modernista site, the Hospital de Sant Pau, which is no more than 1/2 mile from La Sagrada Familia. Open 10-6:30 Mon-Sat and 10-2:30 Sun.) The Miro is closed on Mondays. EDIT: I walked to the Miro today from the Poble Sec Metro station. Over 200 steps were involved, plus several uphill blocks. I'm a stubborn walker, but I probably should have used Rick's information on public transportation to Montjuic.
Casa Lleo Morera, another of the top modernista sites, closed recently for "maintenance", but I've heard that it is not going to re-open to the public.
La Boqueria Market is closed on Sundays.