The first step is probably not in the direction you would expect. Rather than talk to an airline, your first step should be to talk to the people in the embassy of the place you are moving your pet to.
Before you book a flight for your pet find out about any import requirements they must fit in with. The pet movers will know this already, but you need to know quarantine requirements, if your pet can be imported through any port in the country, and the travel papers your pet will require.
After this call your airline. Find out if your pet can fly when you fly, if that's going to work in with quarantine.
Then comes the vet. Pet movers will know about this; almost certainly there will be specialized advice a vet will need to give you. Also, the vet will be able to provide the official documents of health officials will ask for at the other end of the trip.
The day before you leave feed your pet a smaller helping than it is used to. They will travel better on a near-empty stomach. Also take your pet for a walk shortly before flying.
When getting a shipping box ensure there are compartments for food and water, and any medications. But also make sure the size is right. Your pet will want to stretch, stand and turn about.
The most important thing, however, is the documentation. Get it right.
