You dont say how long you have been learning or what you have done so far. But what you describe is a very common problem.
Most riding schools teach students to sit the trot and then to ask for canter.
To make it harder still they ask students to do the sitting trot and ask for canter at the corner so the horse strikes off on the correct lead.
It is harder to balance at a corner.
That is why some teachers like my own teacher teach stiudents walk to canter instead.
Sitting well balance in trot at the corner take practice but ou will do it in the end.
But to get the canter you dont need to kick. Most of us dont kick horses ever!.
The secret of asking a horse to do anything is to prepare first, so the horse is able to do what you ask.
Before I sit to the trot, I make sure the horse is trotting really actively. If you do this extra active trot, with nice contact in the reins, coming up the long side and aproaching the corner of the school, most RS horses will expect to be asked to canter at the corner.
All you need to do then is to relax your hands, touch your legs gently and briefly to the sides of the horse, and allow the horse to move freely forward between your thighs and hands. I often say the word canter too.
The allowing is important.
You have to want to canter - and this is a contradiction as the first time you are not dsure what is going to happen. It is best to learn to canter in a lunge lesson. Then te RI asks the horse to canter and you can learn to sit canter smoothly without worrying about brakes and steering.
I think skib made a good point - are you confident about cantering? Horses are so darned sensitive to our unconscious signals. If you're a bit nervous, they'll pick it up, and you'll be forced to work harder, which in turn could make your legs go all over the place. If you can, keep your heels down, that might stop your foot slipping. Also, yes, saying "canter" could help too. My boy responds quite well to verbal commands.
I wrote a long post - read it carefully. You dont want to squeeze thre sides of a horse.
Just touch and release. Squeezing on both sides makes the horse slow down. Like bbicyle bakes pressing against the wheel.
All four legged animals need to swing their bodies from side to side as they move. This gives a space for the hind leg on that side to step forward. Next time you walk on a horse try to feel that swing from side to side.