The Mezquita has to be one of Europe's spectacular sights. I had been three times before, but I should hate to think that my fourth visit was the last. It is now easy for the carless to visit the wonderful archaeological site of Medina Azahara as there is a return bus trip for €5.
Three other new places to me, all worth a visit, were the brilliant archaeological museum, the botanical gardens, and the 13 patios at the Palacio de Viana. I have done a journal on Cordoba already on this site but this seemed enough to justify another, although the mezquita has probably had sufficient coverage.
Quick Tips:
There is nothing better than just wandering round in Cordoba looking at the patios -- don't be embarrassed about it; they are there to be seen and there is a lot of pride in having them admired. Make sure one of your walks passes through the amazing Plaza de la Corredera.
Best Way To Get Around:
See above for Medina Azahara. I have only walked before, but now, with a dodgy heart, I found the urban bus mightily useful when I was loaded with my stuff.