In Granada the airport is about 30 minutes into town via bus (to Gran Via de Colon). The bus service cost around €3 each and was pretty much timed to accomodate the flights into/out of Granada every day, and I recall it also made an intermediary stop at the train and/or bus station.
In Seville we dropped the car off at the GoldCar Rental, which was at a Shell station near the airport. The agency then took us via shuttle to the airport where we hopped the bus into town. For the life of me I can't remember how much the bus was or how long the ride took (to Avenue del Cid), but it wasn't long at all. It was then about a 10-15 minute walk to our hotel in the Santa Cruz neighborhood (hotel was 3 minute walk to cathedral).
Given we didn't drive into either town, I can't say how the experience would be. I do recall thinking how I would never want to drive IN Granada, as the streets seemed to be a bit chaotic and I believe they have restricted zones prohibiting unauthorized drivers. The bus/train stations seem to be enough on the edge of town to be ok if you are just driving right out of (or just into) town.
The same thing with Seville - the train station appeared to be ok for picking up/dropping off purposes.
But these are assumptions based on watching traffic from the safety of a sidewalk. Hopefully someone else can relate their personal experience with it!