Cadiz, the oldest inhabited city in Europe, is a tiny peninsula on the south coast of Spain and can be divided into two parts - the new town on the 'neck' of the strip of land and the old town at the 'head'. The new town looks like most coastal cities in Spain, with large throughfares lined with palm trees and beaches with bars and clubs catering to the sunburnt tourists. The best beaches are here in the new town, but the old town is where the character is.
As you pass through the cities walls that encompass the old town, the road splinters off into narrow streets that lead into the centre and a couple of minor highways that encompass the old town. It is best to leave your car around here and travel the rest on foot.
Old town Cadiz is a collection of plazas joined by a series of narrow streets. Nominally, the 'main square' would be Plaza San Juan de Dios which is home to the ayuntamiento (council) building or the Plaza de la Cathedral (guess what you'll find there!), but plazas Candelaria, Mina and San Antonio could be considered your focal points too.
Starting from Plaza San Juan de Dios, it is a short walk through the shopping district to the Cathedral. From here, walk up c/Compañia to Plaza Libertad, home to one of the oldest indoor markets in Spain.
Take the street in the northern corner, c/Hospital de Mujeres, to see the women's hospital with its ornate church, and then hop to the next street up, c/Marques del Real Tesoro, to see the Torre Tavira, with its panoramic view of the city. From up there you can decide for yourself where to go next.