My barn owner is freak about the gates, even when the horses are in the barn. The barn is right on a busy main road. So we actually paddlock the gates, and the key is in the tack room.
Stalls are another big thing, they are to be cleaned everyday thoroughly before the horses come in.
Picking up after yourself and your horse (if it poops in the aisle, clean it up etc.)
Being on time and not making the barn owner do your chores for you. (some people get too comfortable in a self care situation, not saying you will, but I've seen it happen)
Make sure you don't try to be cheap and feed your horses less to save money. A lot of barn owners get mad about that, i've seen a lot of self care boarders do it...
Dogs, if you bring your dog, make sure the barn owner is ok with it, and that if the barn owner has dogs, they get along, and you keep yours on a leash at all times, and your dogs don't chase the horses.
If I think of anything else, i'll post it!