Have a flight to Gibraltar and my outbound flight is from Madrid. The outbound flight isn't booked yet, so my date is a little flexible (plus or minus a day). I have no desire to visit Granda (Alhambra) or Seville as I had been there some years ago. I am particularly interested in Corboda (La Mezquitta) and La Ronda if this is feasiable without a rental car. Which bus companies operate from La Linea to Cordoba or am I better off going to Malaga and taking RENFE to Cordoba? Not sure how to fit La Ronda with Gibraltar - Cordoba - Madrid. I plan on spending two nights in Madrid as I'd like to go back to Toledo, so I am looking for a budget hotel near Atocha. Any other "must sees", I don't think I'll be returning to the region soon and would spend a maximum of eight days, not including flights, in the region in November.
'Not sure how to fit La Ronda with Gibraltar - Cordoba - Madrid.' From La Linea, make your way to San Roque or Algeciras by frequent local bus service. From San Roque or Algeciras, take the (scenic) train journey to Ronda. From Ronda take the train to Cordoba (may require a change). From Cordoba take the train to Madrid.
Does this cover it? 'must sees' are subjective - take a bus from Ronda and see a couple of pueblos blancos, the Medina de Azahara very near Cordoba, which many do not see, perhaps.
I really liked the bullring in Ronda. As Neil said, Medinat al-Azhara is very interesting. I saw it with an inexpensive tour. I would not have enjoyed it nearly as much without the tour. You can buy tickets at the Tourist Information kiosks for the round-trip bus excursion, with or without the guided tour. If you haven't been to a horse show, there's a nice one in Cordoba at the Royal Horse Stables. Just across the Romano bridge from the Mezquita is a very interesting small museum. There's a nice view from the roof. I was underwhelmed by the Alcazar - nothing at all like the one in Seville.