I agree with Hoss1975 about the 22LR, I've disposed many a chicken killing canine with my 22LR. If they still enough, I'll go for the head shot, if not, I'll do the wing it heart/lung. Sometimes a double-tap is required---usually not, the 22LR is more then enough for dog sized predators. I have a bolt action 22LR that I've had for years, I also have my fathers 1946 tube fed bolt action 22LR. Both work fine for varmints. My Son has a 17 HMR and I've seen that it's also ample for dog sized varmints.
P.S. By the way, paper targets and varmints are two very different shooting situations, you'll need to practice at both.