LoneStar,
22-250 - probably the most popular and versatile 22 centerfire ever! Great accuracy and good barrel life but with enough velocity to be effective to 400+ yards. Rivaled only by the .243 as an ultra-long range varminter. Ammo is readily available in a vast array of configurations.
.223 - Obviuosly ammo is plentiful since it is the current US military round. Reloading components are lying on the ground at most ranges. VERY accurate when handloaded or with premium ammo. Effective range is limited to about 300 yards max with varmint weight bullets.
.222 - SUPREME accuracy - the QUEEN of the 22 centerfires! It's a shame that the .222 is waning in popularity but effective range limitations in the 200-250 yard area have cost the little cartridge in todays "High-velocity" climate. Factory ammo selection is limited but handloaders can shoot them forever since the low intensity cartridge is easy on brass.
For your primary application, turkey and coon, the .222 would do the job in spades. If you handload and like to have something a little different from everybody else, this is your ticket.
If you want to get the ultra-long range benefit for double duty on prarie dogs or groundhogs, it's the 22-250.
Me? I like the .223! Hell, I can barely see 400 yards much less shoot a squirrel that far. Practice ammo is cheap and you can handload stuff that will shoot sub 1/2MOA.
Mikey