Itinerary Check Please for Spain in Late May

Take the train from malaga to seville as it is most efficient method and for four folks plus luggage you would need a larger car. We base out of malaga and use a car for exploration, but not for point to point travel. Plus ditching the rental car in seville can be a real pain due to navigating the streets.
You need to reserve the tickets for Alhambra NOW! Do your web research on this issue. Hmmmm..... With three nights in each locale highly recommend you rent an apt. Advantage is more space, separate bedrooms, kitchen and we feel a better economic deal. Test the waters by viewing the arbnb website.
Stop in cordoba as a day trip on your way to madrid.
Stop, groan"....... Just reread your post.
Ok, reset. Fly directly to granada as three days here will provide better opportunity to explore according to the "likes" you described. The nearby sierra nevada mountains offer fantastic hiking/biking and should be still slightly snowcapped.
Take train from granada to seville then train from seville to madrid.

I just came back from Spain. With only one week to spare, we focused on the South (Andalucia). My kids (12, 18 & 21) and wife loved it. We landed in Madrid. Toured it for the 1st day and then on the 2nd day rented a car and toured (in Order): Segovia, Toledo, Granada, Ronda, Seville, Cordoba, Madrid .

Was originally planning on doing the coast but shifted to seeing the "White Hill Town". On the drive from Ronda to Seville, took a detour to Grazalema and Zahara and they were lovely small towns. The road very scenic too. For the coast, my different readings pointed to "nothing much" in Malaga, "expensive and ostentatious" in Marbella. Probably sight-seeing the eastern coast off Marbella (and beyond Malaga towards Motril) would be nicer and less commercial. Since you're going in May, I would expect the coast to be "livelier" than in December, so probably worth it.

Mourad

Malaga has more flight options as it is an international gateway, and you could pick up rental car here.
Granada though would still be a preferable starting point in my opinion.
You wouldn't need a car if you were travelling direct to Sevilla - the train is fine.
If however you wish to see some countryside and stop in beautiful Ronda a car would be the best.
Ditch the car in Sevilla though and continue on to Madrid by high speed AVE train.
Cordoba can be a day stop on your way to Madrid - store your luggage at the train or bus station.

Thanks folks for your comments! -

Marbleskies - Will definitely check out the bike riding -- thanks for the suggestion:

Mourad - I love the route that you took! Did you drive this entire route or take some trains? Which towns did you stay overnight in? I appreciate your comments about Costa Del Sol, hitting the coast near Motril might be the way to go, though I've heard so much about Marbella, understanding that's its ostentatious.

djp_syd - will the high speed train from Seville to Madrid allow us to stop on in Cordoba? I was under the impression that it was a direct shot….

Ditch the car in Sevilla though and continue on to Madrid by high speed AVE train.
Cordoba can be a day stop on your way to Madrid - store your luggage at the train or bus station.

You would need to book 2 separate journeys but most services stop in Cordoba.
4-5 hours layover would be enough time to see the Mezquita.
I think Sevilla-Cordoba is 1 hour and Cordoba-Madrid 1.5 hours.
Check up re luggage store as I know this has been an issue in the past.