Ask them what they normally cover in their lessons and see if you can watch a few with different level students, so you can see how they work with each one. It's always helpful to have a talk with them about what your goals are so they can inform you of how they would help you to achieve these goals. Ask them what their expectations of you would be if you became their student. Would they want you riding a certain number of times per week (not necessarily lessons, mind you). For example, when ready to move up to the next level my old trainer would expect us to be riding 5 days a week, with one or two lessons spread out in there. So there would be at least 3 days where I'm riding to practice what I've learned and get/keep my horse in good shape.
Make sure the trainer teaches the type of riding you want to do as well. Seems obvious, but sometimes people forget to make sure the trainer is going to teach them hunters instead of jumpers.
I agree with everything Strange said, and here's my input:
Do take one or two lessons, see how you work together. It might work in theory, but sometimes a teaching style doesn't mesh with how you learn.
Look around the barn, is it cluter free, clean? Do the horses look in good shape and care? Will they make you use their vet and farrier or will you be able to use who you like? Is the tack room organized with enough room for all of your things given? Does the facility cater to what you and your horses want/need?