As always, my advice for the first (jet lag) day is get as close as you can to where you're going. The day is about shot already: you arrive early AM and most Hotels or Hostals won't guarantee a room until mid afternoon. So you might as well keep going (i.e., fly to Sevilla).

You may have some difficulty finding accomodations in the price range you are looking for, especially if you are not a Spanish speaker. When I last visited Spain in June'04 I had already booked Hostals using the Internet (to locate them) and calling them direct to reserve accomodations. Don't use 'third party' services to locate and reserve places to stay. They're in business to make a profit and drive up the cost. The US Dollar's weakness vs. the Euro makes things bad enough. You don't need other people getting their hands in your pockets.
BTW
In June, I made an Año Júbilar Xacobeo visit to Galicia: I stayed 2-nights at a Hostal in VilaGarcia de Arousa (on the coast) and 2-nights in Santiago de Compostela. Double room (single occupancy) in VilaGarcia was 35-Euro /night and the same in Santiago was 30-Euro /night. Each was with private bath and each included (continental) breakfast. The one in VilaGarcia had an elevator; the one in Santiago did not.

I understand your desire to visit Sevilla, Cordoba and Granada (each city has something unique and special) but Malaga and Gibraltar?