Having been through this with two daughters I can tell you it's all about their comfort and the ability for you to teach them to safely control the gun.
If you think it's gonna be a one gun and done thing, think again.
My oldest, whos 19 now but was 7 when she first started shooting, learned trigger discipline, fundamentals and just flat out what to expect when a gun goes off on a P22. Not the gold standard for accuracy but again, it wasn't about shooting 1" groups at 20 yards at first.
As she grew comfortable with shooting she became confident enough to try guns that weren't the best fit in her hand ala Ruger MKII and S&W 41. Because of her confidence with shooting she was able to focus on grip, trigger etc. Within a short time she was handling those like a champ. Today she enjoys 1911s but especially a 686 with some good solid 110 grainers.
My youngest who's 12 right now lasted about 6 months on rimfires and loves her some VP9.
I guess my point is, teach them fundamentals on something that fits them, not a budget or a preconceived idea of what you think should be happening. The worst part of teaching anyone is making them uncomfortable or scared of our sport by not listening to their needs.