.223 would be the cheapest and most common ammo, and the Savage Axis you mentioned, a Ruger American, or Remington 783 would all probably be plenty accurate enough and not expensive.
I have a 22-250 I would sell, pm me if you want to discuss, but it's a Rem 700 so one of the above would probably be less expensive. Ballistics are better though, in the 4000 fps range or just below.
And unless you reload, you can't get 22-250 as cheap as .223, and some places might not stock it (although it's not at all uncommon).
Thanks for the offer. Recoil and cost are my 2 main issues. I have some $ to throw at this but not a bunch. (Gotta do something with this Gov handout. ) Trying to keep it between $400-$500 for the rifle and enough ammo to practice with. Truthfully, this probably isn't going to be for much else besides the coyotes that like to get to close to the house and chickens. Maybe a hog if needed, but we haven't seen them for about a year now. It just needs to be something she can handle since I'm only out there on weekends.