You probably won't ever get "the most mellowed-out equine possible" up to a trot
The only way to learn to ride is to ride. But a few tips:
1) make sure the stirrups are adjusted long enough. You should be able stand in the stirrups on the balls of your feet and just barely clear the saddle with your butt. Otherwise, your knees will be killing you after an hour or so.
2) keep your legs extended (see #1). You should be riding in basically a standing position. This lowers your center of gravity. DON'T grip with your knees and thighs - it will just squirt you up out of the saddle.
3) keep your butt down deep in the saddle. Try to sit on the hip pockets of your jeans instead of just your crotch.
4) learn to move with the horse. Excuse the analogy, but it is kinda like sex
5) keep your hands low. Don't pull on the horses mouth - western horses are trained to ride with a somewhat slack rein. Touch the rein to the opposite side of the horse's neck from the way you want to turn. If he doesn't respond to that, pull slightly on the inside rein, keeping your hand down near your thigh instead of up in the air. RELEASE PRESSURE when he responds. Repeat if necessary.
6) keep your hands off the horn unless you are about to fall off. When you hang on to the horn, you destroy your balance and movement. It's really pretty hard to fall out of a western saddle, as long as you keep one leg down on each side of the horse (see #2 again)
7) last but not least, treat your horse gently. They are a prey animal and have a different mindset than predators like dogs and humans. Hitting is inexcusable unless the horse is being aggressive (like biting) and then only for about 3 seconds after the offense. Horses respond to release of pressure - ask, and then release when he even starts to do what you want.
Hope that helps a bit. Relax, the horse will be tense if you are tense.