Borrowing a goat can be difficult because as others have already posted they are definitely escape artists and LOVE attention. They'll be just like a dog if you allow it, only they won't stay in a fence. If you do decide to borrow a goat, some will leash well. I had two that I put each on a long leash with two swivels in so it wouldn't get twisted up. I tied the leash to a truck tire (transport truck or tractor tire) and put a low water dish in the middle of the tire.
Some horses will make friends with anything. Goats and donkeys are common, but roosters, dogs, cats, and geese are all options too depending on your situ. Since you're moving you want to keep it simple and consider if you can take the new buddy with you.