17 days (October 25 – November 10, 2019) 15 nights--- How to do Andalusia

That's a great amount of time for the area. I'm not sure I'd rent a car for the whole time though, kind of a pain in the cities. I'd take the train from Madrid to Cordoba for one or two nights. A lot of people do it as a day trip but it really has enough to warrant at least one, preferably two, nights. Then take the train to Seville for at least 4 nights. I would start the car rental upon leaving Seville. There are numerous 'white towns' - small villages/towns not far from Seville. Ronda is the largest, most well know. Maybe base there and visit some of the others by day trips. Then to Granada for at least two or three nights. Then back to Madrid. If you still have a day or two extra I'd visit Toledo, either with the car on your way back to Madrid, or via train from Madrid.

Totally agree with Isabel. It is really easy to travel among the main towns by train and bus, and a car is only an asset if you want to go to villages or natural areas. I love driving trips, but I would definitely make use of the fast train to Cordoba. The basics are easy--Sevilla, Cordoba, Granada, so what you should now do is concentrate on the extra that will most captivate you. The Rough Guide to Andalucia is excellent, as is http://www.andalucia.com/.
If you can alter the Madrid booking, we found it easy to just hop on the train upon arrival so all of our Madrid time could go together at the end of the trip.

So here's the big question are you really not interested in the touristy spots, do you really want to get lost and go off the beaten path? If that is the case, then I would probably bypass most of the typical destinations and sights of southern Spain, like Granada, Alhambra, Cordoba, Ronda, maybe even Sevilla.

Instead I would rent a car in Madrid and drive southwest through the Extremadura region, a land of time warped Crusader Castles, Moorish Strongholds, Royal Monasteries, and impressive Roman ruins: stopping in places like Cáceres, Trujillo, Guadalupe Monastery, Mérida, Zafra...

Afterwards, you will find yourselves in Sevilla for a few days, but I would not linger too long, I would continue south to the Doñana National Park, to see the wild nature of Andalucia, staying in the pilgrimage town of El Rocío.

Then continue down the Costa de la Luz, all the way to old port town of Cadiz, a gem of a city. Then on to the lesser known White Hill Towns of the area, like Jimena de la Frontera and Medina Sidonia. Make a u-turn back up to Sevilla, dropping off your car and catching the AVE back to Madrid, and fly out from there.

Carlos's idea is intriguing. Extremadura is very interesting, and a car would be advantageous there.

For the major cities in Andalucia, a car will be a major impediment as well as much, much slower than the AVE trains.

If you embrace Rick Steves, but don't want to hit the touristy spots, then don't buy his Spain book.
Also, please confirm where your return flight is from.
Carlos has some good ideas for lesser known places and this is where a car may come in handy. I know there are a number of gardens, castles and strongholds around Madrid that could easily be a week's road trip. I was there at that time and Madrid needs gloves and a hat as well as warm coat. Further north or higher altitudes may need a light down puffer coat.
Nov 9 is Madrid's patron saint's day which is good for local festivals, parade and seeing how Spanish Catholics put on some pomp and ceremony.
But if you want Andalusia, there are reasons that the cities are prime tourist cities. The Alhambra earns its street cred honestly....and if you want to see it, then you need to book tickets ahead as they are even now very limited for October.
Cordoba offers a lot of Jewish heritage and information - much more than the Rick Steves 'day trip' consideration. I stayed 2 nights and there still wasn't enough time. There is an old Roman excavation and museum just out of town that I couldn't fit in due to being there Sunday and Monday - so limited bus and tourist opening hours for the other sites.

I love that idea, and I will keep it in my back pocket for when I return to Spain! We'll have to see what Florida Wanderlust meant by not-touristy spots, as most still intend to see the "big three" of Andalucia. For the record, I skipped Granada, twice (!), but I was rewarded by Ronda. You might consider choosing one or two of those cities and then using the car to its full potential with a jaunt through the Grazalema area.

I've spent 2 to 3 weeks in Andalucia on each of my 3 trips in the past 8 years. All my trips were in February-March but late October and into November should also be off-season. I traveled mostly by train, rented a car only for the white hill towns, because they aren't easily accessible by public transport. The trains are fast, reliable and comfortable, taking you from city center to city center. You may spend more for trains than a car would cost (including parking and gas), but you'll use up more time driving and looking for parking, plus driving in the cities is a pain.

There are a few minuses that I see in Carlos's suggestions for a first-timer - you'll have to decide how much, if at all, they would impact you. (I have put his itinerary in my files for my next visit . . . after one week in Andalucia I've been hooked on Spain). First is that you would be skipping some major sights - the Alhambra and the Mezquita. Next is that in my experience, outside the major cities (and even in parts of them) English is rarely spoken. In the places with high foreign tourist volume, sights have full explanations in English and some have audio guides. You will probably need to do more research than for a more standard/ordinary itinerary. Lastly, driving on unfamiliar roads and to remote places may not be relaxing.

Madrid - You should be able to get through the airport and to Atocha station (Madrid) by 17.00. That's plenty of time to take the train to Cordoba or Sevilla or Toledo. All are chock-full of history. You need to be in Madrid on your last night to insure getting to the airport in time. You'll need to leave Madrid around noon to be there 3 hours before your flight. If you leave all your Madrid time to the end of the trip, you avoid a hotel change. Sure, you could stay in Sevilla or Cordoba and take an early train to Madrid, but why have the stress of a possible snag that would cause you to miss your flight? If you love European painting, you need two full days to visit the 3 top art museums. For the interests you listed, there's not much in Madrid. The only thing that comes to mind is the Naval Museum (think Columbus, Pizarro, Balboa . . . ).

For the history of the Moors, Christians, Jews, my top picks are Toledo, Cordoba, Sevilla and Granada.

If you want to visit the Alhambra (the #1 sight in Spain for good reason), you should book tickets as soon as you have your dates . . . or conform your itinerary to the available dates. Right now the official website shows tickets are sold out through Nov. 3. Plan on at least 2 nights in Granada to have a full day at the Alhambra. There are still tickets available for the night visit to the Nasrid Palaces Nov. 8. I enjoyed seeing the buildings at night before my daytime visit - they look completely different and there are some views of the city below.

Sevilla is a good base for day trips to Jerez (horse shows at the Royal Equestrian School, sherry bodega tours), Cadiz, Italica (Roman ruins), and possibly Arcos. You can day trip to Cordoba but it is so rich in history and sights that you'll miss a lot without spending a couple of nights there.

I would consider renting a car in Sevilla to spend 2-3 days through the hills (staying in Ronda 2 nights or 1 there and 1 in another) and dropping it when you get to Granada.

Toledo is another good place for 2 nights. It's on a spur line from Madrid, so you have to go through Atocha station to get there. It's a half-hour train trip with frequent trains during the day