To add a little more, all smooth transitions require the horse to engage it's inside hind leg. If horses aren't straight through their back they can't engage their hind leg and find it particularly difficult to strike off from walk or halt. All horses have a natural crookedness and depending on how well schooled your animal is, some horses will lean on their outside shoulder. You can tell if this is happening if you work on the inside track of the school and if the horse drifts out to the outside track (almost imperceptibly rather than the horse making a definite bee-line to the outside track) you have a horse leaning on it's outside shoulder. There are various exercises to correct this, but it's probably easier for you to talk these through with your instructor.
However, one way to correct this: you need to get a very active walk going, put your outside leg behind the girth and keep it there and keep the pressure strong against the horses side. Take up the contact with the outside hand and ask for a little neck bend to the outside. Put your weight down through your inside leg and hip. If you find this difficult turn your head over your outside shoulder - this naturally puts your weight into your inside - and keep it there as you ask for canter. Once you have got the canter and it is going smoothly you can look back to the front!
If your horse is running into the trot form walk, it is likely that he is unbalanced during the transition. Instead of keeping on asking him through the trot, half halt him, try bringing him back to a walk, balancing him up again and ask again.