There is a bus station across the street from the train station which has lockers. Yes, it is safe to use it. I have not heard of any problems with it.
Then take a taxi to the Mezquita/Cathedral. It is not far.
A good place for lunch:
Casa Pepe de la Juderia
Calle Romero, 3
This restaurant is elegant and the waiters wear black and white and are well trained and very attentive. The china is made especially for the restaurant in Limoges. The salmorejo (typical thick tomato soup) and the rabo de toro are very good. For dessert the flan is very special and extremely delicious. There is a small patio which is a good place to eat if it is not too hot.
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