My instructor told me just to relax! You want to use your core as others have said, but try your best to keep your legs relaxed, think about keeping them stretched down to the floor but loose - I tend to try and grip with my knees and then that just makes the bouncing so much worse! If you could get your friend to lunge you on their horse in trot and you cross your stirrups in front of you, just a few mins a time and it will all start to fall in place...that said I still bounce about!!
With canter try to follow the rocking movement with your hips (while staying up right so you don't get left behind). You can then also do this slower and the horse's pace should slow, or do it a bit more actively and it should encourage a more active canter...but this is all about practice too!
I have been riding for 16 years now and still learn something new every ride, it's what makes it so fun :) so stick with it and it'll improve... you'll look back after a year of regular riding and not believe how much better you are!