Close to Austin, Reimers Ranch is a nice spot to go climbing. We did it in a Day trip from Dallas. There are all kind of level and nice river to go to when the climbing day is finished.
To enter the park you need to pay 10$/vehicle.
For the topo : mountainproject Reimers
There is no particular season to go there depending on how you like hot or cold climbing ! We went in February and it was warm enough to enjoy climbing.