Practice, and make sure he's got a steady canter. The faster they go, the more bouncy they are, which is hard to sit to. Forget your stirrups, these won't help. If you're trying to keep your stirrups or are putting too much weight in them then your seat isn't going to be on the saddle, it's going to be in your feet which is prohibitive. So forget your stirrups for the moment, remove them and sit on your bum. Make sure that when he comes back into trot you steady him up as they usually do a fast running trot after canter which is even harder to sit to.
Make sure you sit tall, with your legs long, and with a straight line going from your head, elbows, bum and ankles. If you tip forwards or back when cantering you will start to bounce. If your legs tense then you'll bounce. You need to get a nice steady canter, relax your legs long and low and just try to 'feel' the movement. If you're having to kick to keep him going, that will affect your legs and thus your seat. Make sure he canters promptly from one aid, and that he keeps going without nagging until you ask him to stop.
Have a lunge lesson, so someone else controls the speed and all you have to do it sit and concentrate. You may find it helpful to grab the pommel and pull your bum into the saddle. It's hard to relax your legs if you're boucing, as your natural reflex is to grip, but this just makes you tense and more likely to bounce. If you're bumping about on his back your pony may well rush and get tense also which will make things worse.
You could try standing up in canter, in your jumping position and really concentrate on the canter. Feel how he moves, what his rhythm is like, ask for a faster canter, then slow it down. When he's nice and steady, gently lower yourself into the saddle and join the movement. Book some lessons, they are invaluable.
As for the moutning without stirrups you could try learning to vault on. I'd advise doing this on an immobile object such as a gate or pile of hay before you launch yourself at your pony. Or join a pony club and do the games - they'll teach you vaulting there.
Good luck, keep practising and have fun!