There are two compelling reasons for competing in dressage rather than hunter disciplines: manes and tails. Hunter jumper braids are a form of torture that starts with the actual braiding, continues at the show (where your lack of craftsmanship brings unwanted attention), and culminates in braid removal at the end of a long day. I've never tried to do a tail, but standing behind a horse's rear and and pulling tail hair into a tight weave seems nothing short of foolhardy..
I can trim a tail passably well, and button braids are pretty forgiving. My youngster has soft pliable hair that is easy to braid tightly. Lord knows I'm no expert! My only advice is to read as much as you can, and practice! Here are the resources I've found on the topic.
Looks Like Winner... -
How to make dressage button braids from the Dressage Connection
My personal favorite set of instructions. Good photos, short and clear instructions.
Anky Von Grunsven's button braids
Nice pictures--and if you press CTRL-+ you can enlarge them for a better view. Take a look at this one, if only to see Bonfire with his hair down! The unrolled braid is the nicest and tightest of any in these instructions. However, the final step of sewing the braid up does not look complete. The braid is very long, but the instructions say to turn the brain under only once. Doesn't it look like they've been turned a couple of times?
Button braids on COTH, several posters give photo examples of their braiding plus descriptions of their technique.
Robbie's braiding clinic on COTH
A poster summarizes what she learned at a braiding clinic -- with pix.
Button braids and how tos discussion on the Ultimate Dressage bulletin board
Art of English plaiting
Very nice pictures. This article instructs the braider to "turn under" the completed braid prior to securing with a rubber band or sewing thread. It seems to me that this would loosen the braid, but if the braids are sufficiently tight to start with it probably won't matter.
Button braids by Wiki-how
Instructions for elastic/rubber band and sewn in methods. The instructions are good and very readable with nice, clear pix. but the finished result is not super-polished compare to say, Bonfire's.