Three week first trip to Spain for a senior couple

Spain has a lot to offer and any if the areas offer day trips, so if you want the minimum amount of moves, Madrid, Seville and Barcelona could work.
You could shift things to include a few nights in Granada as well....the Alhambra is amazing.
Toledo is often a day trip fromMadrid, but if you don't mind more moves, one or 2 nights in Toledo allows good visits for a number of secondary churches, museums and sites....and some breaks from the daytrippers.
Segovia is a pleasant day trip from Madrid as well. It was altitude, so will still be nippy, but lovely.
I haven't spent time in Bracelona, but there are numerous day trips. You could even look at Zaragoza for a day trip, or a stop on the train route from Madrid.
Cordoba is a great day trip from Seville, bit also offers enough for a couple of nights.
None are too far apart to make overnights challenging as long as you minimize bags and are mobile.
If art is your thing for Madrid, and you decide to do short trips from the city, staying near Atocha train station and 'Museum Row' would be handy.

Three weeks in Spain...heaven! One week in Madrid is not too much, and if you do day trips from there it may not be enough! We are not art museum people, and we have been to Madrid multiple times. There is so much to do, it's a beautiful city and people are wonderful there. And the food!!! Love Barcelona as well, and a week there would also be good. On this forum it seems most people try to steer visitors to Barcelona or places other than Madrid, so that's why I'm plugging Madrid! Have not been to Seville or Granada, so can't address those. Toledo is lovely and worth a trip as it's so close to Madrid, one day or do an overnight, but I personally don't think it needs more than that. We have done Cordoba and weren't that impressed, but others seem to see something we didn't. Valencia appeals to me, but it doesn't get a lot of attention on here (basically none from Rick), but have not been there. Next time we would probably do several days in Madrid, stop in Zaragoza for a couple of days, a couple of days in Valencia and a few days in Barcelona. We have always taken trains in Spain and never had an issue although depending on where you wind up going, there may not be an easy train trip to get there. Busses are a good alternative for those places. Enjoy your trip, you will love Spain!

Sounds like a great trip.

We spent about 2 weeks in Spain last fall after getting off a cruise ship in Barcelona. We had a couple of days there before the cruise, and a day after... enough time in B. to do the main sites, but not Montserrat. Going back this fall for that.

Then train to Bilbao for 2 nights, then train to Madrid for 4 nights. We were able to see all of the art museums, as well as a day trip (group tour) to Avila & Segovia. Then train to Grenada, Cordoba, & Sevilla; each for two nights. Rented a car in Sevilla for the drive thru the White Towns to Rhonda (stay at the Parador there). Then drove down to the sea, and up to Malaga airport for a midday flight back to Barcelona for one night, and then home to California. Overall, this was a fairly busy trip, with little allowance for bad weather or illness. In town we mostly walked, or got a taxi (when we had the luggage).

Trains are a good way to get between the major cities (that you probably don't want to drive in anyway), and the Renfre website & app facilitated the bookings. With more time we would have included Salalmanca, Valencia, Zaragossa, Merida, Jerez, and Toledo. And more rural rental car driving.

Hope this helps.

Here is what Rick recommends:
https://www.ricksteves.com/europe/spain/itinerary

More important than age is whether there are any mobility issues.

A rental car is a hindrance in the city, so if you only intend to visit the larger cities, take the train or bus. With the time you have you could visit each of the four cities you mentioned. Since I’ve been to 3 of the 4 cities multiple times, I’ll let others provide their idea on the amount of time you need in each. I wouldn’t add any additional places to visit.

If you decided against the main tourist destinations and instead visit smaller cities, towns, and villages, then a rental car would be advantageous.

We did the Dali pilgrimage the week we spent in Barcelona. In doing so, we took the train from Barcelona to Figueres. After touring the museum there, we rented a car ~ picked up at the train station ~ and drove to Cadaques, where we overnighted and then toured the museum in Port Lligat the next morning. We then drove back to Figueres, returned the car to the train station and took the train back to Barcelona. The drive to Cadaques is beautiful, and seeing the landscape there gives further insight into Dali’s work and his inspirations. We stopped in Roses for lunch and a walk along the beach as we were heading back to Figueres. It was also lovely.