Older horses seem to be the best kind, they are kind and patience, although mine is only 12 and great for kids.
Good kids horses are hard to find, mostly because now days horses are boarded out, not worked with daily by riders, and then usually in just a pen. what happened to farms where horses and ponies were rode daily by kids everywhere, and experienced all kids of objects and animals, and machinery. I guess it was different back then....the good old days.
Horse/pony should be temperment tested before buying. If your not experienced, have someone that is test the horse/pony. Do everything you can think of or have seen happen. i.e. bump there back when swinging into the saddle: Do they stand or jump around. Act like your going under their belly. Do they move, kick or stand like a rock. What happens if when you put the saddle on the stirrup "accidently" winds up underneath the saddle. Think like a kid. The best horse I have bought was a 6yr. old Halflinger/Quarter mare. Slightly green. I'm an experienced rider. On my good days, she'll test me. On my bad she's a good girl. Put my very beginner son on her and she doesn't go faster than a walk. Goes where he says. But stops before walking into a tree, and waits for him to tell her which way to go. Best horse I've owned. Take your time finding a horse/pony. Follow your gut.