The nearest airport to the capital is in the town of Jerez, 50 kms away via highway.
It is an international airport, though medium size. From the airport there is NO bus to Cadiz. You can take a taxi to Jerez Center and there take a bus/train to Cadiz. Or take a taxi to Cadiz directly, for about 10 euros
The nearest airport is Jerez, then Sevilla and after Malaga.
there are some cheap companies now dealing with this aerport , for exapmle ryanair flies here from Standstedd
They have a good bus connection from Madrid (6 or 7 hours), trains from Sevilla are very often.
I will suggest to rent a car.