First and foremost, get him responsive to your aids. Don't even worry yourself over the transitions right now, until he will respond immediately to the basic aids.
Start with halt-walk-halts. Ask him to walk off lightly with a touch of your calf and movement of your seat. No reaction, then give him a flick over the backside or behind your leg with a Dressage whip. It shouldn't take him long to figure out that it's easier to go from the light aid.
Same deal for the downwards aids, let him walk, then stop your back and touch the rein for a halt. If he doesn't halt, then a quick 'hey you, wake up!' on the rein will encourage the halt. Again, it won't take him long to learn to respond to the light aid.
You say you are 'slow', is this in terms of your reactions on the horse? Because if your reactions are not lightening fast, you're never going to train a horse to be light. Work on that one!
Now your canter-trot transitions, I believe this is stemming purely from lack of balance and probably lack of confidence on both sides.
If you don't have a balanced canter, you won't get a balanced transition.
Initially I would ride literally millions of walk-trot-halt-trot-walk transitions, get them spot on and staying in balance.
SO many people think downward transition and take their leg off. The leg is there to tell the hind legs to keep coming. Take the leg off and pull the reins, it is the riders fault, not the horse's, that the horse goes splat on it's front legs and loses balance. Especially if you are not blessed with a horse that has it's weight naturally over the hind legs.
Stay on a 20m circle, and aim to ride a transition every 1/4 circle. I don't give a rats backside where the horse's head is, if it looks pretty etc. You just need to be effective, and once you are effective with a reactive horse, it will start to look pretty.
Pick up a loose, balanced trot, leg yield off your outside leg towards a 15m circle, then back out to a 20m circle. Do this a few times until the horse is yielding softly off each leg, remaining in balance.
On yielding back out, at around 18m, slip in your canter aid and continue to ask for leg yield out. Canter for only a few strides, then come back to trot by imagining you are nailing yourself to the ground. Stop your body, engage your core and squeeze your leg on. Keep your body upright, you should not need to use the reins unless he runs through this aid. In that case, a quick upwards 'hey you, listen!' on the outside rein will suffice. As soon as you get back to trot, ride forwards, and continue the leg yield exercise. Asking the trot to be bigger or smaller each 1/4 circle, and sprinkle the canter in here and there. It is so important to keep the hind legs coming through in every transitions, or you will force the horse onto the forehand.
If you allow the canter to come onto the forehand before the transition, you will have a hell of a time trying to ride a good transition.
Even if your horse is not at the level yet, start thinking towards riding canter-walk transitions. Nothing improves your seat in a downward canter transition like canter-walks. You cannot pull the reins in a canter-walk, it must come entirely from your seat, and if you have allowed the canter to get away from you, the transition just won't happen.