At about 6, my daughter started helping me clean my guns. I used the opportunity to: (a) teach her safe gun handling; and (b) teach her a little about how they work. Within a year or two, we took her out with a BB gun. Unfortunately, she didn't follow directions very well, so we had to cut that short. By 8 or so, she was ready to have another go with the BB gun.
The next year, at about age 9, she & I decided that she was ready to move up to a .22LR. We took her out to a buddy's house. My buddy is also an NRA Youth Instructor. He has a suppressed 10/22 that he uses for training, and he let her shoot that. I made myself scarce so as not to interfere with his instruction, and just hung out loading magazines for her. She went through about 300 rounds of .22LR that day. Then my buddy dragged out a suppressed Remington 700 in .300 Whisper that he also uses to teach the young'uns to shoot. THAT was the one she really, REALLY enjoyed. She's a pretty good shot, too.
The next year, she discovered nail polish and hasn't been back to the range.