From the airport, just follow the signs to Cadiz. You will follow the ring road around Sevilla. Then get on the toll road to Cadiz. Near Cadiz, instead of going into the city, follow the signs to Algeciras. At Vejer de la Frontera, head towards the sea to Barbate. It's not really complicated. Once on the road, you can pick up a map at a service station. Or try to pick up a Michelin map at a bookstore, like Barnes & Noble before you go.
I've been dying to get to Barbate. In June we spent a weekend in Conil and I loved it. Barbate is supposed to be spectacular. Have a great time!