Here's what Elizabeth told me we are doing in Cadiz: we're docked for a long time and most things will not be open at 8, so we will probably not rush off the ship. (I suspect the long day will help those who want to spend the say in Seville, though). From where we will be docked, we can walk right into old Cadiz. Plaza Espano is very near the port, and later in the day will have lots of markets with local handicrafts and food set up. She says that there is another area elsewhere with big modern department stores (she bought three dresses at Cortes Ingles) but that's not what we are looking for in one day in Cadiz. She suggests heading to the Torre Tavira first, which looks like it is about 13 blocks from the port on the map she brought home. It has a cool display of an old camera obscura, but the best part is going to the top and seeing all of Cadiz. She says it's easy to get your bearings that way.
Then she suggests going to the Cathdral with its gold dome. It is easy to find, and see, she says, and looks easy walking distance from Torre Tavira. After that we are heading to a beach. She always went to Placa de la Victoria, but it's a little far from the city center (more about transport in a minute). Placa Santa Maria del Mar is a little closer to old Cadiz, and walking distance if you like to walk (maybe a mile or so). There are places to eat near the beach and places to shower off.
She often took buses, which she says are easy -- you just pay the .90euro (90 euro cents, and they will make change -- she came home with a ton of .10 euro coins) on board, no tickets required. Otherwise, she also said that taxis are cheap, although you don't really hail them on the street -- you look for taxi stands which are around in old Cadiz and near the beaches.
She warns everyone that many shops do close for siesta (about 12-4, approximately) so that would be a good time to be at the beach and shop on the way back. Elizabeth says that Cadiz is small and easy to get around on your own, and she liked it a lot.