It is highly unusual to boarding facilities to allow boarders to bring in outside boarders unless the barn owner receives a cut of the lessons, at least in my area. Here's what I look for
1. Well fed horses. I don't want to see ANY horses, even the barn owners', that look too thin.
2. Stalls with ample bedding-- I shouldn't be able to ever see the ground/mat underneath (this is a big part of why I am leaving my current barn). I would NEVER EVER consider a barn where this was a problem, no matter how good the rest of it was, now that I've had bad experiences with it.
3. Communication-- someone in charge should always be on call over the phone or on premisis-- not necessarily the barn owner, but someone who can make the call to call the vet or farrier in an emergency. For non-emergencies, there should be a clear procedure for returning calls/emails.
4. A turnout schedule-- when will my horse be on grass vs. dry lot? If there are several groups of horses, how are they rotated through? I want to KNOW my horse has been out.
5. This one is oddly specific, but you end up seeing how important it is when it doesn't happen-- I want to know that the stablehands feel comfortable contacting the barn owner about issues with the horses. They check in the most on the horses through feeeding, mucking, turnout, etc, so they are in the best position to monitor horses. I want to see good communication between them and the barn owner (another reason why I am leaving current barn-- barn owner heavily discourages stablehands from contacting them, so they are now scared to do so in borderline emergencies. When they have gone to barn owners about horses that refuse to get up or eat, they were ignored because they are "just" stablehands).
6. Full water buckets in stalls at all times.
7. This one isn't a complete deal breaker, but it's important: a webpage detailing what is available in the facility, what boarding entails, and how much everything costs. I'll be much more inclined to look into a barn if basic information is easy to get. Again, this is probably the lowest factor, but it's just good business sense that a lot of horse people seem to ignore for some reason.
8. No surcharge for "extra" hay or grain (unless it's supplements or a truly extraordinary amount way outside the facility's norm). Horses should be given the amount of food they need based on their metabolism and size, end of story. Trying to upcharge for basic necessities comes off as really unprofessional. If it is absolutely necessary to charge extra, the exact amounts of food should be spelled out in the contract.