Using one of the reputable companies is the best way forward the first time you do it. Like most things it's not overly complicated, but if you don't know the steps, it's easy to skip that *one* bit of paper that's needed.
Bear in mind it's not just the paperwork / effort getting them out of Oz, and the various airline regs, but the stuff needed to get them into whichever country you are heading to. If you have a connect, make sure the airline allows access (on tarmac usually) so you can get water to them. I've yet to find an "automatic feeder" that works.
Different airlines have different policies but I do believe that in Oz it is governed by customs rules. And given the bureaucracy of this place probably means you need them at the airport a day before!
We once flew three dogs (German Sheps) AND a cat from SIN to DXB... and checked them in as excess baggage!! But that was SQ and they want to help. Basically the airline didn't care as long as ground handling in DXB didn't care. Simply arrived at check-in with the dogs (and cat) in their IATA approved transport containers and loaded them onto the scales. (Then a guy with a trolley took 'em off our hands - no they did NOT go down the chute!). Getting them off the carousel in DXB was a bit of a bitch however!
Of course when we flew to SYD from DXB (same animals) it was a whole different problem as SYD wouldn't accept them as excess - we HAD to check them in as cargo. No real difference except they spend a bunch more time at the airport before the flight, and you spend a bunch more time trying to find them at the receiving end
Of note was that my wife (flying 'J' on the DXB SYD route) could hear the pups barking for the first few hours and the last few hours of the flight - she thinks the hold was under First Class... so that would have been *really* loud in the leather seats .
