Eek. Im going to start in western pleasure, but I am horrible about bouncing. Trotting is the worst, but cantering is hard too, because some say its okay to bounce, but 'scoop' your seat, and others say just sit and don't bounce.
What do I do to cure the bounce? Besides posting, LOL
I'm not really sure about what's correct or incorrect for western pleasure, as I've never done it.
But for not bouncing -
Just relax and breath. Sit tall, sink you're weight down, and use you're lower half to sort of "flow" with the horse.
Practice, practice, practice. Just get out there and keep working on it. =]
The world is best viewed through the ears of a horse.
Make sure your horse is striding out low and long, maybe some trot poles at a distance slightly longer than his normal trot stride to make him stretch out more and not go as vertical! Then you can ride the smooth trot without bounce ;)
And the same for the canter, make him nice and long, without speed, for a smoother canter. Use canter poles slightly longer than his regular stride to encourage him to stretch out and not up as much.
Make sure to loosen your hips and let your lower body move with the horse. Your upper body is going to move a little, just remain loose and relaxed. Don't tense up if you're moving too much, which is really hard (experience speaking ).
What's worked for me, which may not be the right way, is to just keep trying to sit the trot until I learn the movement of the horse and how to move with him. He recently started giving me a canter, so I have no advice for that yet.
"Sit tall in the saddle, hold your head up high. Keep your eyes fixed where the trail meets the sky. And live like you ain't afraid to die...don't be scared, jut enjoy the ride." - Chris LeDoux
Some horses are just plain smooth and it's easy to sit the trot and canter - it's all in the angle of the shoulder and pastern. The canter should typically be a smoother gait then the trot and easier to sit.
When I ride one that is choppy, as mentioned above, I'll work harder at sitting deeper and getting into rhythm with my horse. I'll also collect him and that smooths him out as well.
I'm not arguing with you, I'm just explaining why I'm right.
Nothing sucks more than that moment during an argument when you realize you're wrong. It's not always what you say but what they hear.
I don't know much about Western Pleasure, I ride English and bareback, but I will offer my advice anyways.
I would say to do stirrupless work and really follow your horse. Maybe some bareback riding, too. I would just keep on practicing and do not tense up! That will just make you brace and give you no hope of "melting" (as Solon described earlier ;) ) with your horse.