Well, for any livestock, at least in my opinion, the question is how close can you get. If you can walk right up to the goat, pig, cow then in my opinion the best place to hit them is in the brain stem. That being said, not many will let you that close. The traditional is to shoot right between the eyes, slightly elevated. The behind the ear at angle is also a great suggestion, that works well, but again if you can't get close to them it isn't workable, so the between the eyes shot is the necessary one.
A .22 is the work horse of any ranch, often being used to put down livestock.
P.S. EDIT
A pheonix arms .22, while an admirable little weapon, has no place for this kind of work. Go to the store and buy a beater .22 rifle new or used and it will do fine. I recently picked up an old Glenfield (marlin) model 60 for $45.00. After cleaning it, which apparently had never been done since the gun was bought, it shoots like new.
Another edit...
My grandfather was very fond of .38 specials for this kind of work as well. Another underrated "working" gun on a farm.