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Depends on the size of varmint. I prefer a .22 Hornet... good for 150 metre shots on varmints up to the size of a fox. Closer range it will cleanly and reliably take a coyote. Others will say that you "need" a 223. And yes, you're in the right forum.
For coyotes I use 243, any 22 center fire for smaller stuff.You don't list where you are, but in Alberta we sometimes get a kick at a wolf. More gun is better for them.(ie:243 vs 223)