I used to volunteer and ride at Kingsweston Riding Stables by Blaise Castle in Kingsweston and they are brilliant.
They're a privately owned stables and the owner, Richard Slaughter is wonderful. The helpers are all really nice, and they have wonderful confidence-giving horses and ponies.
It's a very adult-oriented environment, even though a lot of their students are children. Some people have been riding there 10 or more years!
If you need any more help, feel free to contact me! I'm right by Kingsweston and would be happy to meet you there for a hack to get you used to the place.