First question would be to see if they can solve your current problem. Say your current problem is lack of turnout - ask how long the horses are out.
You can ask any question you can think of when you are inquiring about boarding. I have had some really really really 'out there' questions.
One thing I will advise is to not talk badly about your current barn. All the new barn needs to know is that the situation was not a fit.
Basics are feed, water, shelter, turnout, hours, areas on property to work the horse, what is included with board and what do you need to provide. Of course price.