Picking a riding stable is a big decision. Most likely, you will be riding and taking lessons there for a long period of time.

You should do lots of research. Look in the phone book or at local tack shops for riding stables in the area. Make sure this is a drive you can do at least once a week, don't pick a farm too far away from you, or you'll be regretting the drive with the gas prices lately.

Make sure you stop by the farm on a day before you lesson to visit and see the condition of the barn and horses. There should be a decent shelter in each field, or stalls for the horses to come inside. The horses should be bright eyed and at a good weight. You should not be able to see the backbones and ribs of the horses, and if you do, run the other way and call animal control.

Talk to some of the other people that ride at the barn. See how they like the lessons and the trainer. Watch some of the lessons and see if you like how to lessons go. Make sure you can deal with the trainer's personality.

Ask the trainer any questions you have, and do not let anything go unsaid. Feel free to ask me questions as well!

Edited: For those of you who wanted to know what types of questions I'd ask, here you go.

1.) How long have you been riding?
2.) How long have you been teaching?
3.) WHY do you teach?
4.) What is your philosophy when working with horses and students?
5.) What are you prices for lessons?

Those are some very general questions, but let me know if you'd like more in-depth answers.