-Nice-sized arena, good trainer(s), nice-looking place, nice horses, people are nice, etc. & a reasonable price.
When I'm checking out a barn, in general...it must be in good condition, the horses are well-cared for, the people are nice, & the pricing has to be good.
Also, an indoor arena is a plus. & good trails, as I loveee to trailride on nice warm days! Get info about lesson times too of course lol.
Always make sure the horses are well-cared for (your horse is going to be one of them!) & also sign an agreement/warranty of course. Ask a lot of questions about feed, the farrier, & the schedule of feeding, turnout, blanketing, etc. ;) Or if they use the combined system (which is half turnout half stables). Also if other people are going to be riding your horse (like for lessons). Good conditions (of pastures/fields, tack room, barn/stable, stalls, hay, etc.). You want the best for your horse, lol. Also a good price. & ask about when you can see your horse, like some barns have restrictions on when you can go to the barn (like late at night & such), etc. There's more but I'll add later.
You're definitely on the right track!!
It is very important when you're looking for a place to board your horse(s), the current horses are in good health & cared for properly.
I think I covered most of it, that's my 2 cents! Probably forgot some things, soo I'm going to have to add more soon. LOL!