A 22Long Rifle round should be sufficient. I don't know how they do it now, but 30-40 years ago they used 22 long rifle rounds to dispatch cattle in the slaughter house. It should be all you need for goats. But as mentioned by 007sean previously, the most important thing is where and how accurately you place the round. Which should not be a problem, as you should be within inches of your target. and recoil should not be an issue with this round.
