Website for purchasing tickets: NOTE - tickets from Madrid-Cadiz are not yet available (they will be by the end of May): http://www.renfe.com/EN/viajeros/index.html
To get to Cádiz by train, you should go to Atocha Station (same name for metro station and train station) and then go to the area Puerta de Atocha, where the high speed train departures.
To get to Atocha Station there are two options:
- Metro (see subway map below)
- Bus (see bus schedule below)
Train Schedule: Madrid to Cádiz Cádiz to Madrid
These schedules are approximate, and June schedules will be published online by the end of May. Check schedules before travelling at http://www.renfe.com/EN/viajeros/index.html
MADRID AIRPORT to TRAIN = BUS SCHEDULE