Private tour recommendations in Barcelona?

You'll need time-specific tickets to the individual sites in order to skip the often-hellacious ticket lines. It's easy to buy those online ahead of time. I suppose a private guide could arrange those tickets for you, but even with tickets you may get held up at the door, waiting for some people to exit; that's how crowded a lot of the most popular sites are. And conditions inside are so crowded I'm not sure most of those sites allow outside guides to conduct tours. The crowding may slow you down substantially, and I wouldn't be thrilled about paying for a guide's time when the sites have audio guides available.

I took the tourist office's Picasso Tour in 2016, and it was perfectly fine, providing a combination of walking tour and museum visit, but the guide couldn't do anything about the then-disastrous crowding inside the museum. We also waited at least 10 minutes for him to pick up our tickets. But that tour cost only about 22 euros, so the wasted time wasn't all that significant to me.

The Palau de la Musica Catalana offers its own tours, which are mandatory at some times of the year. There are multiple English-language tours each day, but they can sell out, so your tour should be booked at least a bit ahead of time. Some of the other top spots may have their own tours as well; you can check the individual websites to see.

I want to emphasize how crowded it is likely to be at La Sagrada Familia, Parc Guell, Casa Mila/La Pedrera and Casa Batllo, as well as at the aforementioned Picasso Museum. If anyone in your party likes to take photos, you'll need to allow extra time so he/she can set up shots with sightlines not blocked by a bunch of other visitors. That will not work so well if you're part of a public tour, and even on a private tour you might have a problem getting to everything you want to see.

Given the situation with the coronavirus, I would not be in a hurry to book a guide without a generous cancelation policy, nor would I buy sightseeing tickets very far ahead. The exception to the latter would be the Camp Nou Experience, a soccer-related thing that I believe sells out months in advance.

I really liked the Barri Gotic tour conducted by the tourist office. The T.O. limits the number of people on its tours. The "free" Barri Gotic tour groups I saw in the same area were a lot larger. I would recommend using the T.O. tour for that part of Barcelona if the timing works for you and spending your private-guide money elsewhere.

Edited to add: There's also a company that does public Civil War tours in Barcelona (about 25 euros per person, and the tour lasts close to 4 hours). This is an excellent tour for people interested in that period of 20th-century Spanish history. Understand that it is heavily information-sharing rather than visual, because the Civil War battles took place elsewhere. The day I took the tour (August 2016), there were only about 6 tour members. I'm sure they'd set up a private tour for you.

Full Disclosure - We did not use Spanish Trails for a walking tour in Barcelona, but we did take 3 small-group tours and one private tour with them back in 2017. A couple of the guides mentioned that they also did some of the walking tours. All of the guides were excellent, and I would definitely recommend this company:

https://spanish-trails.com/category_tour/barcelona-city-tours/

Thank you both so much! Very helpful.
I will look into the tours you each suggested.

Does anyone have info on Barcelona Day Tours? They seem reasonable and have many options. Just wondering if anyone has had experience using them?