We will be in Spain 4/14-4/23 We plan going directly South to Sevilla , Granada and Cordoba first avoiding some Holy week hub bub and then back to Madrid and Barcelona. I am trying to figure out the best transportation around the South if we do not rent a car
I'll leave the transportation to somebody else. You should know, however, that Semana Santa is a bigger deal in Andalucia than it is in Madrid, especially in Sevilla and Cordoba. Plus the core areas of these two cities (the only areas you'll probably be interested in) are relatively small and will be the site of all the action. You'll have more wiggle-room in Madrid.
All is very accessible by train and bus. However, the big issue is time and distance. Madrid, Cordoba, Sevilla, Granada makes for reasonable circle in your time frame. Barcelona because of the distance from Madrid or Granada is difficult to include unless you drop Sevilla. You need to look at a map. Also need to consider an open jaw ticket so save time avoid back tracking. With ten days it is difficult to do more than 3 location changes and you are trying to do five.
...run mainly on the plain.
My apologies...I just had to!