I started out with BB guns then .22lr and shotguns. My grandpa had some land and briefly taught us how to shoot and be somewhat safe. He would get us a 500rd box of .22 and tell us to have fun, there was three of us at age ~13 running around shooting snapping turtles in the pond. With my child, we started out with a .22 rifle and she hasn't shown much interest since, but I have recently shown her how to chamber a round (snap cap) in a 9mm pistol and that your index finger is the primary safety, I want her to know how to use it in case she ever needs to use it. I just wish she had more of an interest in shooting.
I was started at 5 with a Red Ryder, by 7 I was shooting a 22 my grandmother had and at 9 I bought my first 22 rifle. She taught me gun safety and how to shoot young, and we had a blast during the summer shooting cherries off the cherry tree at 25 yards.
With my kids, grandkids and other's kids;
at 3 yo, or whatever their age first time I had them. I filled a 2 liter bottle completely full of water and sealed lid tight. Showing them an assortment of cartridges, .22LR, .45, .30-06, 12 ga. Talking safety and responsibility. Kneeling down on their level. Load a .22 LR and shoot the 2 liter at 5 yards or less.
Depending on the kid, usually 6 or 7 before handling anything other than BB gun on their own.
By 9 yo, they are hunting with shotgun and deer rifle. Going to Cowboy Action shoots with me.
Talk to them with respect. Give them responsibility. Expect them to be responsible. Most will be responsible.
Invite their parents. Even they don't shoot, seeing the responsibility of the kids and the fun they are having, will have a positive impact.