If you don't want to drive there is a direct express bus from the airport to Estepona (great spot), and Marbella. About a tenner for the journey. Timings can be found on ALSA bus website.
I think Benalmadena or Fuengirola might suit you. It is close to the airport, easy access to Malaga city which is great for a day out, and plenty to see and do around Ben and Fuen, it is made for holidays, obviously! I mention the bus in case you might want to go further West.
If you stay in Ben or Fuen you can take the train to Malaga, and from there all points fan out. You don't need a car to see Seville, Granada, Cordoba, Ronda - all can be done by train or bus, although check the journey times to see if you will have enough time for a day trip. Train can also be taken to Caminito del Rey, El Chorro station but it's not for the faint hearted, if you want to take a look online!
Andalucia is great. Hope it all works out for you.