Getting to Cadiz can be a bit complicated, and you will most likely have to make at least 1 connection, somewhere. If you happen to be in Spain already, you may be able to get a flight to the nearby Jerez Airport, but most likely you will have to rely on other modes of transportation.
Flights
If you're traveling to Cadiz, most likely you will have to fly into another city and then transfer to land travel. The nearest airports to fly into are Seville, Granada, Malaga and Madrid. The least expensive flights go into Malaga or Madrid, but the travel time after landing is considerably longer. On the other hand, while it's usually more expensive to travel to Granada or Seville, you will have less travel time after landing.
Trains
Once you've landed, 1 option is to take the train. This can be more costly than the bus but travel time is usually halved and comfort levels are extremely high. There are 2 daily trains from Madrid--a morning train at 10 a.m. and another at 4 p.m--with each being a 5 hour trip. From Malaga the trip is nearly 6 hours with 1 transfer, but 4 trains run between 7:45 a.m. and 4:30 p.m. There are only 4 trains from Granada, and a transfer somewhere is required. The trip takes at least 6 hours, depending upon transfer time. The train trip is only 2 hours from Seville, and the trains run regularly all day.
Buses
There are regular buses from all of the major cities to Cadiz, with costs being considerably lower than the train, but travel times are increased by a few hours.
Cars
You can choose to rent a vehicle in any city and make the trip yourself, but be aware that it can be nearly impossible to find a rental car during the busy season of July through August.
Boats
Alternatively, you can choose to fly to Tangier, Morocco and make the approximately 2-hour ferry ride across to Cadiz, with a stop in Tarifa.