A couple years ago, there was a varmint problem at my parents place, and I was enlisted. Well, I more volunteered.
Anyhow, I was using a RWS 34, shooting a .22 cal pellet at ~800 FPS. Skunks and rabbits went down pretty easy. I don't remember any that needed
a second shot. The one coon I dispatched, well, that was another story. It took around a half a dozen shots to put him down. Looking back, the airgun was not up to the task.
You probably don't need a really expensive gun. Gamo has some models in the right power range for about half the cost of an RWS ($170-250). But they don't have much in the way of .22, if that's what you want. .177 should be up to the job, but I would probably stay at 900 fps minimum, 1000 being preferable.
Crossman also has a few models that could work, for even less than the Gamo ones.