As long as you aren't in the show pen, posting the trot in western tack is fine. In most shows, if you look in the warmup pen most if not all of the western trainers and exhibitors will warm up in a good forward rising trot. I know a lot of endurance/long distance riders will rise or 2-point the trot regardless of tack to take advantage of a more ground-covering stride.
That being said, there's something to be said for sitting the trot even on a bouncy horse. The big tips for sitting the trot are first, don't try unless the horse is round and offering her back to you. If you sit while she's inverted, she'll be rougher because of her posture, making you bounce more. Add to this the fact that you'll be bouncing on top of a posture that isn't conducive to comfortably handling that weight, and she'll invert further in self defense. You'll have a vicious cycle of inversion and bouncing. There are lots of threads on this forum and articles/books out there on getting the horse properly round from back to front.
Next, once you've got a round horse at the rising trot, the most important thing to remember about sitting the trot is to RELAX. Every joint needs to be passively accepting the motion: ankles, knees, hip-joints, waist... even tension in the little joints in your toes can contribute to bounce. Feel your hips matching the movement of the horse's hips; 1-2, trot-trot, left-right... You're trotting together, not sitting on a trotting horse.
All of the above is applicable to the canter. It's harder to ride in harmony with an inverted horse, creating a vicious cycle. That controlled relaxation is still key, and you're goal is to canter with the horse, not sit on a cantering horse. The biggest difference is the motion of the hips, and there are a few mental images that I've found helpful. I like to imagine the motion of skipping/mimicking a canter on the ground, feeling that lead and the lifting circling motion. Another mental image that helps some people is to think about hula-hooping. Be relaxed, but not sloppy - if you want your horse to have self-carriage, you need to have it, too.
Hopefully that was helpful to you. Ideally, you'd probably benefit most from some lungeline lessons to focus on the motion and relaxing. Something else I highly recommend, if you haven't read it already, is reading Centered Riding, by Sally Swift. Amazon
.com: Centered Riding (A Trafalgar Square..."> Amazon.com: Centered Riding (A Trafalgar Square...
There's a lot of great info that can really prime you mentally for what you want to do in the saddle to keep yourself and your horse in harmony and balance.