I would recommend asking any questions you may have. About the facility, trainers, training system, boarding, financial things, etc. I really wouldn't say there are specific things you have to ask because every rider and situation is different. But make sure that the facility is everything your looking for. Big turn out, certain type of arena, training style, make sure its "perfect". I also like to look at the horses that are already at the facility because thats most likely how your horse is going to turn out or act in that barn.
I hope this helped!