We started our son when he turned 2, and for Christmas at 2.5 he got his own Marlin 60 and Savage Mark II (cut down stocks on each) Shooting from a bench, front and rear bags, with an integral laser and red dot. At 3, I upgraded his red dot and started him shooting standing with only a front support, by 4 he was shooting pop cans at 50yrds offhand using shooting sticks (tent poles). He's about to turn 7, and stalks rabbits in the woods with a 22 or single shot 410 while I'm cutting firewood. He's shot my ARs and my PRS match rifles benched and prone out to 800 and 1,000yrds, but he's still struggling with how to use the scopes properly (lots of shadow and parallax issues, since we don't have a stock which fits him properly right now).
I ordered a Defiance Ruckus and a Manners compact stock for him for Christmas, and a 6 Dasher prefit from Proof as his first centerfire rifle.
With my Sis's oldest girl took to shooting a little later. She didn't enjoy it until she was 4, also 7 now, and wanting an AR of her own. She's a lefty, so Uncle VT is building her a Stag to keep the port away from her face.
I've instructed a lot of new shooters, and a lot of kids among them. It's really a matter of the kid. Some are ready to deliver the responsibility younger than others. Some are still unable even when much, much older.