.223 would be best cost wise. Younger shooters need lots of practice (We all do) and the lower costs of shooting .223 would allow you do do more of it.
As already pointed out there are newer great hunting projectiles for the .223 that make it a good choice for small to medium game.
I too am a fan of the 22-250 but it takes 6-8 grains more of powder to reload. It adds up if you shoot a bunch.