Intinerary Barcelona/Madrid

That sounds like a terrific itinerary.
As a Jewish traveler, I often feel like an outsider as I tour the great Cathedrals and museums of Europe.
Although I am not especially religious, I was very moved by the sense of connection I felt when visiting the medieval synagogues of Spain.
Of course the big figure in Cordoba is Maimonides. There's a terrific BBC radio program about him that I recommend.
The Mezquita of Cordoba is entirely worth the journey- I hope you enjoy your trip!

Yes I agree Spain has a powerful and fascinating Jewish heritage, that unfortunately is often overlooked by we locals.

I agree that Toledo should be high on your list, I'd even stay a night or two there, allowing you to experience its medieval ambience to the fullest with little to no tourists in the evening hours.

Cordoba is especially noted for being the home town of famous Jewish philosopher Maimonides, I know just south of the city is an ancient Jewish necropolis and in Granada they have a Sephardic museum and a number of Judeo-Spanish restaurants.

Consider visit to Monserratt from Barcelona. ALso, from Madrid, consider day trip to Segovia.

I know you are active seniors but this trip seems rushed to me. Are you coming from North America? If so, I would be concerned about jetlag when you arrive in Barcelona. Do you think you can handle a bike tour at 10AM? You are spending very little time in Barcelona. Personally, I think Barcelona deserves 5 or 6 nights. There is so much to see and do, and you might get very frustrated with so little time. As much as I love Girona, I personally would delete Girona from this trip and spend that day in Barcelona.

I think it is preferable to spend a night or two in Toledo and in Cordoba. These cities are so crowded during the day with tour buses, especially Toledo, but they are so charming and wonderful at night when the daytrippers have left, the cities are illuminated, and more Spanish families are out in the evening enjoying family time. However, if you think that is too much moving around, then seeing both Toledo and Cordoba as day trips from Madrid is better than not seeing them at all. You will love both places. Cordoba, especially, is one of my favorites, and the Mezquita is amazing!!! Take time to stroll around the Juderia and walk across the Roman Bridge.

I appreciate the concern but we have never staggered through the first day on our prior 3 European trips. We prefer to keep
Moving the first day. We stretched it a little 3 years back by scheduling an evening opera followed by light dinner in Lviv but still stayed awake and really enjoyed it. No problems until the day after we got back, always worse west-bound. There is so much of Europe we want to see and so few opportunities that we have really enjoyed self designed “highlights” trips. But I acknowledge it’s not for everyone.

Hm, you have a lot of day trips to places that seem very important to your interests!
I would stay in Córdoba for sure, as follows:

1. Arrive Barcelona
2. Barcelona
3. Gerona - sleep Barcelona
4. Train from Barcelona to Córdoba
5. Córdoba
6. Córdoba to Madrid
7. Toledo - sleep Madrid
8. Madrid
9. Depart

Sure, it adds a hotel stay, but it will feel smoother.
And yes, 3 nights in Barcelona including the day trip to Girona is less than what most would recommend, but I do not find it that bad. You could skip the Picasso museum, by the way - it is an interesting museum but it will not stand the comparison with what you will see in Madrid after.

As Balso indicated, Girona is better as a day trip from Barcelona. The Jewish area is small but the Jewish museum is very good, worth an hour or two. I might even suggest you skip Girona since you don't have a lot of time and the Jewish sights/history in Cordoba and Toledo are much more extensive and better preserved. Also there are tons of things to see in Barcelona. You seem to be real art lovers, so consider a visit to MNAC - a wonderful art museum. You'll see more of the Palau de la Musica on a tour than at a concert. If you want a concert, look at the Barcelona TI website , especially the "buy tickets" link. There are usually guitar concerts around town. Nothing quite like classical Spanish guitar.

Then train to Cordoba for at least one night. It's a 2-hour train ride from Madrid so it's a lousy day trip. The Juderia is the best-preserved I've seen so far in Spain. The Casa de Sefarad is very interesting. There's also a folk museum in the Torre De Calahorra which is across the Roman bridge from the Mezquita. It focuses on the relations between Christians, Moors and Jews. There are also good views from the roof of the tower. In the evenings the Juderia shops close and the day trippers are gone, the area is quietly romantic, with lots of great places for dinner and maybe even some live flamenco music around dinner time - remember that in Spain dinner time is from about 9.30 to 11 or later. This is the best place (on your itinerary) to look for a flamenco dance performance, since Andalucia is the home of flamenco.

Then train to Toledo for an overnight. You can leave you bags at Atocha station, where you have to change trains anyway, and take a small overnight bag to Toledo.

Then spend your last nights in Madrid.