Have you checked the regulations both going into and back out of New Zealand? I know New Zealand is very picky about foreign wildlife coming into their country, so make sure you`ve checked all the rules! Luckily cockatiels are not affected by CITES, that makes things easier.
...are you from the US? If you are, you need to be aware that birds originally from the US are eligable to do an at-home quarantine when you return. To do this your bird needs a form of ID, such as a leg band or a microchip. If your bird isn`t closed-banded, I would recommend finding an avian vet that`s comfortable with microchipping cockatiels. Birds without a band or chip will be quarantined by the USDA for 30 days upon your return, at YOUR expense.
Be careful when choosing airlines. Some airlines will allow you to bring birds on the flight with you as `cabin pets.` Others will require the birds to be shipped as cargo. Birds that`re shipped as cargo are subject to restrictions on the times of year in which it`s okay to ship them. If it`s too hot or too cold it`s dangerous to ship the animal and thereby not permitted. So I would call the airlines, find out their policies on cabin pets, find out the months where shipping animals to New Zealand is prohibited, and buy tickets accordingly. (If you bring them as a cabin pet, the restrictions on shippings don`t apply, of course).
Many airlines will claim that only `parakeets` are allowed in the cabin. `Parakeet` is simply a word that means `long tailed parrot.` So your cockatiel is, in fact, a parakeet. You may want to get the vet to call explicitely call your bird a parakeet somewhere on the health certificate. Maybe he could write the sentence "Cockatiels are also known by the name of (make something up) parakeets." or whater you want. At any rate, I`ve never known anyone with a cabin pet reservation be turned away from bringing their cockatiel, dispite the `parakeets only` restriction.
Flight suits are straight out, by the way. You will required to keep your bird in a carrier for the flight. I recommend a hard-sided cat carrier. Check with the airline to learn the measurements of the space under your chair where you`ll be placing your bird to make sure you buy a properly sized carrier. For the flight itself, please remember to bring a towel to drape over the cage if your bird starts to scream, for the sanity of all the other passengers on the plane with you.
Keep all toys to the sides of the carrier. Any hanging from the center as hazardous as they make hit your bird in the head.
Make sure that you line the carrier with something that`ll help your bird get a grip on the ground and keep her balance.
I would put a millet spray in there for the trip, to give your bird something to eat. Water can be added in a very small amount that won`t spill or (perhaps your better option) you can put a very watery fruit such as a slice of orange in the carrier to keep your bird hydrated.
Goodluck!