If this horse has no soundness problems (and I doubt that he has), he is just telling you to kiss his royal arse. The stopping is an even worse bad habit to get started than the kicking out. These are completely unacceptable forms of resistance. A quick spanking should have been applied the very first time and there would not have been a second time.
Now, I think you will have to get after him a lot more if he has gotten by with this for a while. The longer a bad habit like this has to get established, the harder it is to break, but more importantly, the horse may get 'ill' about doing anything.
You can use a crop on his butt -- preferably the side that he usually kicks out with. BUT, do not just tap him. Spank him hard the very first lick. You do not want to toughen his resolve but you want him to give it up quickly and get the heck outa there. A big round pen would be the best place to do this the very first time in case he decides to bolt. If he does, just ride it out for two or three laps and stay out of his mouth. Then, gradually pull him up and a couple of laps later, ask him to canter again. Then go out and ask in the open or a big ring and quit on a good note -- only when you have asked him to transition down.
I, personally, do not use a crop for things like this. I use a bridle with long split reins made of heavy harness leather. I use an 'over and under' action where I can spank the horse on the 'left - right - left - right' sides very quickly. This keeps a horse from 'ducking around' from spanking it on one side with a crop. You want the horse to go forward, quickly. You can steady and refine the transition later, Right now you need obedience before the horse ups the ante and just starts 'sulling' and refusing to go forward at all.
I have ridden 2 and 3 year olds for 50 years and have seen no ill effects from training them at that age. I usually waited on Arabians until they were 30 months or older, but all others I preferred to start at two. Trotting and cantering a three year old should not be over-working him at all as long as you ride within reason. Obviously, you don't drill on one for two hours.
I think you have to fix this right away before it grows like a cancer. You want to establish a good work ethic and a willing compliant attitude on your first rides. When you put up with this kind of resistance, you create a horse that hates to work and barely does anything you want.
When I trained for the public, I got in several horses each year that had tried crap like this when their owner tried to start them. Even though I got all of them over it, some I know would have been much better horses had they not gotten a more positive start.