I waited until my son could recite Cooper's "4 Rules" with only minor prompting before I put him behind a real gun. He was around 6yrs old when he got that part down. To drive the point home I shot a water melon at close range with my AR to emphasize the "never point a gun at anything you dont want to destroy" rule.
From there we started with a suppressed 22lr so that all he had to focus on was listening to me and practicing fundamentals such as sight picture, trigger pull, and follow through (this was the hardest one to teach, every shot he wanted to pick his head up and see if he "got a bullseye"). I let him gain confidence with the 22 to the point he can reliably shoot clay pigeons standing off hand at 25-30yds consistently without missing.
Since then (he's 9 now) I have let him shoot my 556 AR, and my 6.5 Creedmoor match rifle, both are suppressed but still pretty loud. He's to the point where hitting 100yds with the 22lr is relatively easy with a front rest and if I give him a little help setting up, he has hit our 10" gong with the Creedmoor at 400yds a few times. Other than those few shots he has taken with fullsize rifles, he pretty much sticks with his 22, but whenever we shoot together I'll still have other guns out (some suppressed some unsuppressed) so that he gets used to the sounds and feeling of being around full power rounds. I picked up a really nasty flinch as a kid even though I started shooting when I was almost a teenager (Im talking to the point that I missed more than one deer inside of 100yds kinda flinch). My uncle took me to a crowded public range with a tin roof over the firing line, and had me touch off a few rounds with his Pre64 M70 in 270win as my first fullsize rifle larger than a 22. I probably didnt shake that flinch until well into my 20's.
The main problem for younger shooters is that most guns even if they are "youth" models, or have been adjusted for the correct length are still really heavy/long thanks to the ATF's arbitrary designation that a barrel shorter than 16" makes a gun into an evil deadly assault weapon machine. Theres only so much you can do to balance that when the gun is 3x longer than your arms. My son is solidly 50th percentile on height/weight for his age, and he has trouble with anything bigger than his 22 as far as being able to hold it steady off hand standing. So anything bigger we have to shoot prone or off a bench. Finding stocks that were short enough when he was younger was also a little bit of a chore. The magpul stock on his 22 was a touch long when he was 6, but I'm probably going to start adding spacers in soon for him now.