Regarding the jig-jogging (breaking the walk) - what we did with my friend's track-trained but unraced horse (who practically didn't walk under saddle when she came in, just jig-jogged, trotted and galloped) was made her walk nice and slowly, with regular verbal reassurance and praise, both on the line and under saddle. As soon as she started to jig-jog, I'd make her stop - in the saddle, this involved the verbal "whoa" cue we'd taught her on the line, plus sitting very deeply and with weight back.
Anyway, the following stuff is what I've learned from instructors and from riding TBs, but not an expert and there are much more knowledgeable HF members out there!
OTTBs are very in tune with weight/seat cues. 50kg jockeys are up in high stirrups on 500kg superfit racehorses without any contact except through the hands, on a horse that has been trained to lean on the bit to balance rather than give to pressure, and has little in the way of a lateral mouth. Shifts in weight, and repetitive but not constant hand cues, is how they control them - if you watch, they aren't just legged on as decoration, they are there to guide the horse through the field and control the speeds. A little drop into the left stirrup and the horse will shift left, lean forward with forward hands and they the horse will stretch out and increase their speed. Weight pushed back with the shoulders on the line of the ankle and the ankle-knee line perpendicular to the ground, hands closed and drawn inward and toward the navel, and the horse will begin to slow - most young TBs are not particularly well balanced with a rider aboard, so need to shift their weight back under the rider's centre of gravity to regain that balance, which of course means a shortening of stride and decrease in speed.
If I were you, I would use this training they come with, along with voice cues taught on the ground, to your advantage. Make sure you are sitting super straight with shoulders back, but don't get into a tug of war with the reins. When you want to transition down, think "stop", sit still and deep, shoulders back and open and squeeze and release your hands until you get the desired result. Same with when he breaks the trot - as soon as he breaks without you asking (make sure you're not accidentally asking by putting your weight forward or providing any pressure with your lower leg, some of them are pretty sensitive to leg cues as they're not used to them) bring him back to walk. He's been taught to go unless asked NOT to, now you have to teach him not to go unless you ASK him
It's probably easier for your trainer to show you how to get a canter happening when you want, however - I'm not sure how much of a 3-beat gait your horse has. Star has learned to canter finally but will still break into a gallop or a disunited canter sometimes if not asked correctly or if she feels unbalanced.
You may have to support his front end for a while, til he develops balance under saddle, but do spend some time each session (best if it's towards the end when your horse is a little less excited) getting a nice walk on a loose rein. At the moment, Star will go nicely on a loose rein at the walk, although she's still learning balance at the faster gaits so we're working (with great tips from HF peeps!) on getting her to hold herself up there.
Anyway, hope that helps!