When I film our youngsters, before they are under saddle, Junior stands them up (perfectly groomed and with a well-fitted halter) while I walk slowly around filming from all angles. I try to remember to zoom in on front legs.
Then, I film them on the longe line. I don't film them free because it almost never shows good movement. They either tear around like idiots, or do the BIG TROT with head, tail and KNEES up. We move them in both directions at walk, trot and canter, just a couple of laps each gait.
Finally, we send them through the jump chute a few times, usually the first time they have done it. As soon as I get about 3 satisfactory passes, we quit.
For horses under saddle, same scenario, except movement and jumping are now done with a rider. If I have a show video, I will use that, but if not, a couple of courses at home, and a couple of laps of the ring is plenty. If you aren't interested or knowledgeable enough to see something you like, then this isn't the horse for you.
Never music, and just basic scene changes. Nothing fancy, just takes away from what you want to see. Cut to the chase! I normally use a Flip video camera, and their basic software has served me well.