Barcelona, Madrid and Bilbao

Bilbao (my hometown)...Hotel Gran Bilbao is about 10min uphill from our beautiful old quarter, the Ribera food market and San Antón church. Guggenheim is a must, but also lots of other sights. Market day in Gernika is on Mondays, the biggest and most important day on the last Monday of October (thousands and thousands of people visiting...forget the car, take the train or bus). Quaint Mundaka, Bermeo, San Juan de Gaztelugatxe, Gernika and the Casa de Juntas, Butrón castle...all very much visiting and close to Bilbao.

There's no reason to drive from Madrid to Bilbao: it is quite long, the train takes about the same time.
If you do drive, Pedraza and Burgos are good sightseeing stops, but Segovia is not on the way and deserves a full day: best done as a separate day trip from Madrid if you have the time.
I would argue against driving however.

As for Toledo vs Segovia : I prefer Segovia because the aqueduct is really unique, but Toledo arguably has more sights.

This is going to be a challenge to plan. If you have 9 nights, you do not have 10 days. You have 8 days and some possibly jetlagged/sleep-deprived hours on your arrival day. That means you have just two full days in Barcelona. If you have an early-afternoon train to Madrid, you may be able to squeeze in one indoor sight that morning, ideally something that opens before 10 AM.

I usually recommend 4 full days in Barcelona and certainly no fewer than 3 because there are so many sights, many essentially requiring prepurchased, timed tickets to avoid dreadful ticket lines and possible sell-outs. I can see you are aware the timing will be tricky. Of the places you mention, you should buy all of these tickets in advance: La Sagrada Familia, Parc Guell, Casa Mila and the Picasso Museum. The #1 strategy here is to make full use of your three mornings in Barcelona by scheduling one of those four tricky sights at opening time each day. That way, you won't worry about having to finish up on time somewhere else to make your scheduled entry time.

I'd tend to put Casa Mila on your last morning in the city because I think there's a clearer limit to how much time you will spend there. I hit just over 2 hours, I think, and I'm sure most people spend less time there. If you figure a maximum of 2-1/2 hours inside the place I think you will surely be totally safe. Choose the opening time slot (9 AM) so the place will perhaps be a bit less crowded and you won't have to wait too long at the door. But they probably won't let the entire 9 AM crowd in at the same time.

Do you want to ascend one of the towers at La Sagrada Familia? I did not, and I don't know how much time to allow for that. I think I was inside the church for about 2 hours. I am not much for photography, nor am I at all religious, so I'm sure some visitors stay longer. People have commented on perceived good times for the stained glass windows, but they were certainly lovely for me during my first-time-slot visit in August 2016. Since I didn't have to queue to ascend a tower, I had a good bit of time in the church before it began to get crowded as later groups entered. If you don't want to ascend a tower, I recommend the earliest time slot. Although much of the beauty of the church is above head level, so crowding doesn't spoil the experience, it's still nicer when there aren't so many people there. If you do plan to see one of the towers, I'm not sure there's all that much advantage to trying for the first time slot; it would matter whether they are still automatically assigning you a tower time that's just 15 minutes after your entry time, giving you essentially no time to wander around the church until after the tower (at which point several additional groups of people will have entered.

The Picasso Museum was unbelievably, borderline-dangerously, overcrowded during my 2016 visit. If you've seen recent photos of the nightmarish crowding in the Vatican Museums, the Picasso Museum was similar. It was quite difficult to get close enough to an artwork to read the label posted on the wall beside it. Do not go to this museum unless you are a serious fan of Picasso. There are many other art museums in Barcelona that are much more pleasant to visit. Should you choose to go, you may well find the crowding seriously slows you down. On the other hand, it could send you screaming for the street within 15 minutes. Hard to predict. Obviously, the first time slot would be a very good idea here, because you'd walk into a museum with no holdovers from previous entry groups.

Running out of room. Will continue in a separate post.

Continuing...

Being outdoors, Parc Guell is probably easier to appreciate under crowded conditions and doesn't have to be the first activity of the day--which is a good thing, because you have 4 problem sites and only 3 mornings to squeeze thm into. There is one thing about PG, though. There's a small building there, I think called the "Gatekeeper's Cottage" for which you have to line up for entry. The line was 45 minutes ling by the time I noticed it, and I think I had the second entry time of the day. I haven't read that seeing that cottage is a must. If you do want to see it, you should take a look at the line when you enter the park and make a strategic decision as to whether it seems prudent to join the line immediately. Seeing the cottage ill clearly add substantially to the time needed, if only becaue of time spent standing in the line. Also note that the pay-to-enter section of the park, called the "Monumental Zone", is only part of the whole thing. I have only been to the paid section and have no earthly idea how much extra time one might spend in the free area. I guess I spent about 2 hours in the Monumental Zone, without seeing the cottage. I felt a bit rushed.

Tickets to Parc Guell now include a shuttle bus from (and I assume also back to) a nearby Metro station. That should be faster than walking the mile back to the Metro, but it does kind of depend on how frequently the shuttle bus runs.

You have no time to leave the city. For future reference, Andorra is too far away to be a viable day-trip, nor is it (meaning the capital of the country) a very attractive place. The Pyrenees area is very nice, but Andorra wasn't great in 1972 and has since, reportedly, become more like an outlet mall (tax-free shopping). I have not been to Montserrat, but it takes much less time to get there, and many folks like to take that day-trip. Few of them, I would imagine, are spending only 3 nights in the city. There are many other possibilities in Catalunya, including my favorite, Girona, but it's better to spend some nights in Girona and use it as a base for visiting points north and west rather that making a bunch of day-trips from Barcelona. But this is all for a future trip. You're going to have your hands full with Barcelona.

I really liked the Barri Gotic walking tour offered by the tourist office, but I think I took a 10 AM tour. That would conflict with the idea of scheduling a problem sight early every morning. I don't know whether that tour is offered at additional times.

I'm not a big fan of Madrid except the art museums. It's perfectly OK, a lively place, but from an architectural standpoint Spain has many, many more interesting cities. Toledo is certainly among them. Toledo is actually worth multiple nights, which you do not have. Segovia is also worthwhile but doesn't have quite as much to see as Toledo. If you're serious about the three big art museums, they'll keep you busy for the better part of two days, leaving no time at all for a side-trip. If you are thinking about going to only the Prado, and only because you think you ought to, you can probably spend one of your two full days in Madrid going to either Toledo or Segovia.

I like Bilbao a lot, much more than San Sebastian. Bilbao is less touristy and has a larger historic district, It's very pleasant to meander between the Guggenheim and the Casco Viejo (not a short walk). Are you flying back home from Bilbao, or must you travel back to Barcelona or Madrid for your final night? Especially if you are not returning from Bilbao, I would really question including it on this trip. There's more to see in the Basque Country than Bilbao, and you have substantial travel time going and coming--close to 3/4 of a day in each direction by the time you check out of your hotel, get to the train station or car-rental agency, etc. That's a lot of transportation time for not much time actually in Bilbao.

I am flying home from Bilbao. Living in Boise, I have a LOT of Basque friends that tell me I can't go to Spain and not see the Basque Country. :)

Thanks for the advice! We were planning to drive to Bilbao from Madrid since we thought it would be better to have a car to explore that area of Spain, and since my friend is flying home from Madrid...he will drive back and return the car there. When we priced out train, bus and flight tickets to Bilbao, it was at least $50 each way per person. Our rental car total for 4 days was 153 euro.

The Basque countryside is really lovely, so there's that. I do love the area, but I spent a lot more time there and wouldn't go for just a quick visit when it requires cutting Barcelona so short. To each his/her own.