Remember to relax your corse muscles as when these are tense they will make you bounce around. Also put your shoulders back and roll them slightly to absorb the bounciness of your horse. No stirrup work is good but some horses temperaments aren't calm enough to allow you to practice safely so do be careful!
Lots of no stirrup work! And let loose of your hips and let the horse carry you , also sit up straight!
Remember to move your hips, carry your hands, keep your heels down, and keep your posture! Do your best to absorb the shock. I found it easier to start without stirrups. It’s easier to absorb the shock because the don’t have the tension of your legs with the stirrups.
Ask you horse to round up and collect so that they're using their hindquarters more because that makes it a lot easier to sit to. Also sit to the trot without your stirrups because you are engaging your back, seat and stomach muscles to keep yourself in the saddle.
U do learn to do sitting trot before u canter cause the transition will be 1000 times better that's obvious ur instructor is thinking ahead as in a dressage test for example u will get more marks for doing sitting trot and preparing u and ur horse in a better and smoother way for the transition to canter. I would defenitly start with no stirrups for a short amount of time and then lengthen the time without them each ride have good posture and all weight down to ur heels and if in doubt ask ur instructor that's wat they are there for.😀It takes time but u will get there
Try to go with the motion gently instead of being a sack of potato’s along for the ride (aka me). The slower they’re going the easier it is. And you can sometimes gently play with your reins a bit until they lower their head into a better headset (if that’s a problem) and it’ll make the ride a little smoother. Sorry if my descriptions aren’t great lol
Fyi - It's very very hard to sit trot to a horse that isn't trotting properly. (Unless they're super smooth) once a horse is moving correctly over his back it's 100x easier. Idk why your instructor wants you to sit trot before cantering, doesn't seem smart to me.
Keep in rythm with your horse with your pelvis - rotate and move it up and down and side to side as each step is taken. Sit deep in the saddle and have a firm calf. Keep your head up and make sure you're sitting vertical.
Make sure you have good posture and grip with your inner thighs, not your knees. Sorry I can’t help more because you probably already know these things! My horse is also very bouncy but I guess I just got used to it and learned how to sit deeper. Sorry! Wish I could tell you more! Good luck!