Having just gone through this, I'll see if I can help you out.
Things to look for:
1. Healthy horses. It's ok if one or two are a little underweight, as they could be new to their owner and bringing them along, but if every single horse is ribby, that's probably a problem with the feeding regime at the barn.
2. A boarding contract that protects both you and the barn owner. Ask for a copy to take home and look over.
3. If they're going to be feeding your horse their food, look at it closely and ask for a detailed list of whats included, including how much hay so you'll know if you need to provide more on your own.
4. What's included in the price, turn out and in, feeding, stall and paddock cleaning, blankets on and off, fly masks on and off etc. This will largely depend on what you're looking for re: self, semi or full board, stall or paddock/pasture board.
5. What are their requirements for vaccinations/worming.
6. If it's important to you, do they allow your coach to come to the barn and teach you (assuming you have one already)
7. What facilities are included, again depending on what you need. If you're primarily a trail rider...obviously you want a place close to trails. If their are arena's, are there times when they are not available for use...such as when a trainer/instructor needs them.
8. Inspect the turnout for footing, fencing, water access.
9. Inquire as to whether your horse will be turned out alone or with a group, if with a group, how big, and how will they introduce him.
10. Tack storage, is there any, is it secure, heated, etc.
That's all I can think of for now.