I would wait tell she is at least eight. I really think that the child should be aware how lethal a hand gun or rifle is before being taught to shoot.
A 3yoa is not to young to give some very basic safety lessens to and many a 5yoa has been taken out shooting with close supervision.
My daughter was shooting BB guns at three, .22s at five. At fourteen she was an NRA junior instructor in pistol, rifle, and shotgun.
7-8 is about right for the first .22. I was delayed with my girls due to moves and work so didn’t get around to it until they were 8 and 10. By then they could handle the handgun as well as the rifle. Got’em some .223 time as well. Good day.
Gotta do that again soon.