I look for:
how well thehorses are cared for first and foremost. That they are being fed right aka don't look skin and bones. They are treated respectfully- as in not harshly nor overly spoiled.
Another thing: atmosphere. This is your personal choice. I prefer a quieter lesson barn. WHile I dont really have any experience with show barns, I just think they would be focused more on things I am not too concerned about-maybe i am wrong. But I like to work with my horse in a place that isn't overly packed or busy. Sometimes it does get lonely at the barn I am at, but i'd feel too crowded if there were several people there all the time. But just ask yourself- are you interested in socialization as well? SOme people may not be a fan of lesson barns that have a lot of younger kids, or some may want a place with more boarders...ect. Lesson days at my place are busier, and sometimes I had lessons that had a few more people in it that I didn't care for, but it wasn't constant.
Also, the instructor- broken down into 2 catagories
Does the barn focus on what you eventually want to get into? Mine is more dressage/pleasure riding. They don't do any jumping. So, if I wanted to jump, this barn would help with the basics, but if I ever wanted to jump, i'd have to find another one. The instructor just flat out wants to focus on other riding abilities rather than jumping ( plus, she doesn't want to invest in jumping horses since hers haven't been jumped/worked out to be able to do any significant jumping).
and then, you can take direction from the instructor. Basically you like the way they teach. Not everyone can be the best teacher to every student- everyone learns differently. If you can observe a few lessons, you could probably see what their way of teaching is and see if it will fit your learning abilities.
But in the end, if you go to one and it doesn't work out, you can always look for another. Of course, prices may help determine your choice and location.