Are you riding English, or Western? If Western, be sure your stirrups aren't too short. That seems to be a common mistake and one I made a first. A much more experienced rider than me - my husband - asked me one day before we rode out to stand up tall as I could in the stirrups - my straddle cleared about 6 inches from the seat. He laughed like a loon... Then let my stirrups out, asked me to stand again - this time I only cleared a couple of inches, if that. It felt way too long to me, but he insisted THAT is where you want to be. A couple of rides later, I was used to it and now I immediately know when stirrups are even one hole too short for me. You'll learn too, in time, if you're riding Western.
Also, are you talking about bouncing at a walk or a trot? A lope?
Personally, I have had no lessons, I'm learning hands-on, but watching a lot of videos on youtube. One of the best pieces of advice I've gleaned out of it is that the more tense you are, the more you bounce - especially if your stirrups are too short and your leg stiffens.
You CAN stand up just a little in the stirrups, or you can sit the trot. I'm working on sitting it and not grabbing wildly for the horn because its hard to consciously make yourself relax in the seat, therefore - I bounce. So it takes consciously reminding yourself (for a while) to relax into the seat, get those heels down, don't let yourself fall forward in the seat.
HorseF is right about riding without stirrups. Even just sitting bareback on a laid back non-reactive horse, out in the pasture, with nothing but a halter and a lead is also helpful. Let the horse do whatever (so long as you're not getting hurt) and just concentrate on how the horse moves, what muscles to engage when in your legs and core, but also, again, if Western, learn to be relaxed in body, but alert in mind. NOT as easy as it sounds. If you own the horse and have it on your own property, even better - I'm learning it helps build trust in both of you to just hang out and chill with and on the horse, not asking anything of it, from time to time.
HF is also right about time in the saddle. That's when the real work starts, as I'm learning now. I've asked a dozen experienced horse people, they all say the same thing: Put miles on the horse, miles in the saddle and you'll both get better.