I want to put together a little deal re: shipping with photos and details like phone numbers, but in a nutshell we paid $300 to ship. But the price is dependent on weight (the container plus the animal, distance and dimensions of shipping container. But, I am no airline expert - the following is what we experienced:
AkAir will not let you ship goats (or any livestock as I understand it) as a pet - ie the $150 standard addition to check a kennel with your luggage. I don't know about other airlines. The goats have to go cargo, have a state of Alaska import permit number, meet the space guidelines - be able to stand, lie down etc, have a NON slip surface (not the plastic in the bottom of a kennel), have food and water bowls and a vet check (you may need other certifications depending on the age of your goat - everything is simpler if its under 6 months old). They only allowed one per container (you could do 2 but they had to have a partition seperating them and still meet the space requirements). Cargo is basically 22 cents a pound but somehow they figure in the distance and the size too. The cargo goes as Animal Express and must have reservations on specific flights (they only allow so many animals per flight) and some flights have very limited room (ie we checked into flying out of Eugene and only 2 were allowed so we didn't want them split up so we went out of Portland).
So, based on all the above - your charges could be quite different depending on distance and size...the best thing to do is talk with the airline you will be using - call the cargo department, and it was helpful to actually call the cargo people where we were shipping out of. But, we often got different answers depending who we talked to - they gave us a ballpark figure but the actual charges weren't figured until we checked them in and they were officially weighed. AkAir will let you ship COD if you want.
Anyway - the AkAir people were great - they loved the goats! We flew the same flight as them - so they were waiting for us at the cargo dock by the time we got off the plane and over to it, and a phone message was waiting for us at home -so they were on the ball and making sure our goats were going to be taken care of. We tried to book the easiest/fastest flights - the choice was portland/seattle/fairbanks or portland/anchorage/fairbanks and we chose to avoid seattle as we've had a lot of delays there recently. Luckily, all the flights went smoothly and here we are - goat farmers at last!