60 days in Spain

You should maybe say if you are also flying out of Barcelona. Also, what is the minimum length of stay you would consider? Off the top of my head, Hubs to consider--Barcelona, Girona, Valencia, Madrid, Sevilla, something on Costa Del Sol. You specifically mentioned Andalusia. You could day trip from Sevilla to Cordoba, but I sure wouldn't. I love Cordoba and it was especially beautiful and peaceful when the day trippers left. There is also plenty to see for at least two days. You might also consider some time in the white villages. you could day trip to Arcos from Sevilla and/or Ronda from Costa del Sol. I think the other white villages are too far from any one central spot. The number one destination for Andalusia is the Alhambra in Granada. I don't think Granada is a great day trip from anywhere, personally. I would want to be at the Alhambra by 9 am or 10, and you should plan on a day. Granada has other interesting sites. For me, two days was enough, however. You might also consider San Sebastian or Bilbao in the Basque area as a hub, except you'd want to go there earlier in your trip since the weather there becomes less favorable in the fall.

I love Spain. You are lucky to have such a chunk of time!

I'd leave Andalucia to the end because it is the hottest (and driest) part of Europe.

There are day trips from the major cities like Barcelona, Madrid and Sevilla, but most of them are better as overnight visits, some are worth several nights. Spain is big, which means that it takes a lot of time to get from place to place. Will you have a car for part of the time? If you want to explore small towns and villages, you'll need a car.

I would certainly consider spending a couple of weeks in Barcelona, but no more that a week in Sevilla or Madrid (and Madrid for a week only if you want to spend days enjoying European paintings in the wonderful art museums).

I agree with Chani, there are many places in Spain with more culture and history than Madrid. However, Madrid is a great location for a variety of day trips. I did a day trip I to Segovia which I enjoyed and worked out well. Many do day trips to Toledo. I personally really enjoyed staying over night in Toledo. There is plenty to see and do and it tends to be quite crowded with day trippers during the day.

Nice to have a chunk of time.
Weather may dictate direction....as well as your departure city, but by early November, Madrid can be quite chilly. If the Northern coast has any interest, then start there. As so much of what you mention is in the south, I expect it will be hard to put that on hold until later in the travel, but it is reallllly hot there still. If you are returning via Barcelona, then heading to Madrid or North is a natural option.
It's tough to balance 'don't want to pack and unpack' with 'enjoy the beauty'. You have the luxury of time so those day trips people talk about can benefit from 1 - 3 night stays. You will be cutting yourself short if you only limit yourself to day trips. The costs of all those return fares and having to leave the city before you have seen and experienced it is a poor balance of some packing....in my opinion.
In my day dream life, I have a road trip pencilled out to spend a week or so driving between the small castle and garden towns around Spain. The windmills of La Mancha, the country palaces and homes with their public gardens. It would mean living out of a suitcase for some days but it could be balanced with more extended periods of time in one location.

Oh, so if driving is even a remote possibility, take the time now to get an international driver's license which is required to drive a car in Spain.
Even if you don't want to drive, Toledo and Cordoba are 2 immediate cities that are considered day trips but have much more to offer.

I don't always understand when people talk about all the time it takes to change locations. Yes, it does take some. But, for a day trip, you still have to pack up some things, get yourself to a train station, wait, take the train and then get from the train station to generally city center. And at the end of the day you have to get back. Now, I REALLY plan my trips, some might say obsessively, haha, so my inns are all predetermined and I have directions/routes to get to them. We pack really light with packing cubes so it doesn't take long at all to pack things back up.

Don't get me wrong, I do like when the situation merits, to stay put in a location and feel like a local, and even when I plan a more "moving" adventure I do try to plan routes to avoid one and even two day stays.

I can’t think of any one or two places that would work for the timeframe you’ll be there. Barcelona and Madrid are obvious choices for 7-10 days each (day trips included). However, there will come a point when there are no more nearby locations, and you’ll be spending more time getting to and from locations than actually visiting them. Andalucia is huge and the sights are spread out. Much of Spain’s beauty is away from the cities, between the small towns, and off the beaten path; not in my opinion, along the Costa del Sol. Consider renting a car for a portion of your trip. It’s easy looking up things to see in the major tourist hubs, but’s takes a lot of time to find some seldom visited castles, monasteries, nature preserves, chapels, etc.

I found plenty to keep me busy for about two days in Granada, then I added extra time to do a day-trip to Priego de Cordoba and another to the area around Bubion (great for walkers).

In northern Andalucia I stayed in Ubeda wnd day-tripped to Baeza and Jaen.

With a car you could combine Cuenca, Teruel and Albarracin, but those are not day-trippable from each other by public transportation.