I had the same dilemma this year with my 12 year old son. While I already have a Ruger 10/22, and he has already shot it before, I cannot recommend any semiauto as a first rifle for kids. I bought him a stainless Marlin single shot, and he loves it. It teaches him some shooting discipline, which I did not really learn when my dad gave me the 10/22 years ago.
I totally agfree with other posts that recommend the Ruger on its merits of value and reliability, however, and once you are satisfied that your girls have learned the basics there is nothing wrong with getting a 10/22 as well.
After all, you DO have TWO girls!