Which are the must-see spots in Barcelona that require advance booking?

The "Monumental Zone" of Park Guell.

After you're done with the Sagrada Familia, I highly recommend taking the 10 minute walk to the Saint Pau Modernista site. It's fascinating and wonderfully uncrowded.

Palau de la Musica Catalana, tour or performance;
Casa Batllo;
Casa Mila;
Picasso Museum;
Dali museum in Figueres.

The only other book-ahead place I can think of in Barcelona is Camp Nou (soccer experience). That's the one that probably sells out way in advance.

Sant Pau, aside from being lovely, is very handy because--so far--no one has reported encountering a significant line there. So it's something you can slot in sort of on the spur of the moment if you have time (though it is pretty close to La Sagrada Familia).

What will be a bit of a challenge for you is figuring out how far apart to make the reservations. These are not super-quick places to visit. The tour of Palau de la Musica Catalana is a bit shorter, but the rest may well be 90+ minutes each. They were for me, and I did not ascend one of the La Sagrada Familia towers.

Casa Mila and Casa Batllo are both open late; that will help.

The Picasso Museum can be unbelievably, miserably crowded. It was that way for me in August 2016. If you aren't big fans of Picasso, I'd recommend skipping it. There are lots of other museums in Barcelona--art and otherwise, and the Picasso is the only one that regularly gets packed. Buying a ticket in advance is necessary to avoid a long ticket line, but it will not help with overcrowded galleries. Casa Mila and Casa Batllo also are extremely busy (night hours may be better), but it was still possible to enjoy the houses.

Six of the sites listed above are on the list because of their modernista architecture. I love that stuff, but that doesn't mean every tourist needs to go to 5 modernista buildings and a modernista park. There are lots of other things to do in Barcelona.

If you have any interest in walking tours we enjoyed the Runner Bean free tours. The Gothic Quarter and Gaudi Tours were a great introduction to the sites already mentioned. The guides were knowledgeable, passionate and entertaining. runnerbeantours.com

Casa Batllo and Casa Mila are open in to the evening. If you plan a late afternoon or early evening visit then I think prebooking would not be required.

Big YES for ...
•Sagrada Familia
•Park Guell
•Palau de la Música Catalana
•Picasso Museum

Thanks for the feedback! So let me see if I got it right:

We have concert tickets for Palau de la Musica, so that saves us one line-up for tickets as we won´t do any extra tour.

My husband doesn´t care for Picasso and I saw the museum 30 years ago so we most likely will skip it. But he may want to see the Dali museum (while I don´t care about Dali ...), so better make reservations there and figure out how to get there.

Our apartment is very close to the Sant Pau Modernista so we should be able to do that when nothing else is up, or did I misunderstand this?

My daughter (the trip is to celebrate her graduation as a civil engineer) and I are big into architecture. All the modernista sites are on the list for us, so bookings especially for Casa Mila and Casa Batllo, from what I hear.

Hmmm, just looked a bit more at the Dali museum in Figueres. That would be a full day-trip from Barcelona considering the distance, wouldn´t it? Any suggestion as to how to make that work?

There are fast trains on the Barcelona-Figueres route. They take 55 minutes each way and go to the Figueres Vilafant station, which I believe is a bit over one mile from the Dali Theatre and Museum. Others have reported that there is bus service from Vilafant to the museum.

I didn't go to the museum myself (also not a Dali fan), but I regret that now. Friends who did go said it was fun, and there's a substantial jewelry exhibition that I would definitely have enjoyed. I believe it's on permanent display.

Although I would hate to cram it into the same day-trip as the Dali Museum, your train will stop at Girona as well. Girona has a large medieval district, a wall you can walk on, two interesting old churches (with English audioguides available) and a good museum (art, bits of sculpture from churches, etc.)

We used Spanish Trails for 4 different small-group tours. They offer this tour that includes the Dali museum in Figures.

http://spanish-trails.com/?vh_accommodation=culture-art-tours/dali-museum-medieval-girona-day-trip

Beatrix, a few additional comments:
1. At the Sant Pau Modernista hospital complex, I signed up for and enjoyed an English language tour, which are given at a few specific times, for a few euros. The tour involved just a guide, a Belgian couple, and me. The tour went to various buildings over maybe 90 minutes, and then I wandered around the buildings and grounds for another 60 to 90 minutes on my own. Very interesting, especially as the complex was endowed specifically as a hospital for the poor.
2. Another site you might enjoy is Palau Guell, which is downtown. It is an earlier work of Gaudi, and while lovely and impressive, it was quite austere and darker, lots of dark wood, iron work, paneling and stairways. It was the home of the Guell family for about 10 years, and then I understand even they decided it was too dark, and they moved out near Park Guell. If you go, don’t miss the roof, which is quite whimsical. It seemed like maybe Gaudi followed Guell’s desires for the house, then had a free hand on the roof. Two staff there recommended Tapas Guell for lunch, almost across the street, which was very good.
3. You can visit the Dali Museum in Figueres in part of a day, or combine a morning there with an afternoon/early evening in the old town of Girona, as I did. The fast trains make this easy. The one thing I did which I would avoid was to walk all the way from the Girona train station to the old town, then back. Quite time-consuming and tiring, the walk was not worth those investments, I think. Discover a local bus or take a taxi, at least one way.