Good on you for getting on with some dressage lessons
Collection does not come until both you and the horse are in balance, you have all your transitions spot on and all working gaits are established with the horse over its back and swinging, off the forehand and responsive to all aids.
You're not going to get collection as soon as you start 'dressage', in the beginning you will be getting the most important part down pat... your position, balance and feel. From there, you will learn to start fine tuning the horse and how it travels and eventually you will start to learn more about how to shift a horse's weight back over it's hind legs and begin to enter the start of collection.
As for your trot-canter transitions. It's all about feel, balance and timing, along with the horse's education. Obviously a young green horse is not going to jump into canter beautifully in balance, it is going to run a little and swing it's body. However I'd say if you're riding a lesson horse, it would be balanced into canter, it just depends on how it is ridden in- horses are simple and lazy creatures, if the rider doesn't ask for maximum energy output, horse won't give it haha!
You need to have a nice steady, soft balanced trot and a good solid sit trot. Go from sit trot to canter, keeping your outside rein, inside leg and supporting with the outside leg to stop the quarters from swinging to the outside in the transition. I'm sure if it's a lesson horse, it will very well know the aid for canter, so it's more a case of you learning how to control his way of going until you can tune in enough to each other to work out the best timing for an aid and how much aid to apply.