This is what I would do in your position.
1. In the UK, departing for Australia:
(a) Checking in to the flight: show your NZ passport
(b) At UK exit passport control (if the desks are staffed): show your British passport
(c) At the plane door: show your NZ passport
2. On arrival in Australia, at Australian entry passport control, show your NZ passport.
3. Travelling between Australia and NZ (in either direction) you need only use your NZ passport.
4. In either Australia or NZ, departing for the UK (if you ever make a trip back):
(a) Checking in to the flight: show your British passport
(b) At any security check: doesn't matter, but easiest to use your NZ passport, probably
(c) At NZ/Australian exit passport control: show your NZ passport
(d) At the plane door: show your British passport
5. On arrival in the UK, at UK entry passport control, show your British passport.
This is because the purpose of the passport check varies:
- • Airlines check your passport to make sure that the name matches the name on the ticket but primarily to ensure that you have the correct papers to be admitted to your destination country.
• The security check on departure is just that: if they ask for your passport it is to ensure that the passport and the boarding pass match.
• The official passport check on leaving a country is to make sure that you are not an illegal immigrant or a wanted criminal (basically) and to record your departure (in the case of Australia and NZ, which keep such records).
• The check at the plane door is just the airline making a last-minute check about documentation again.
• The official passport check on arrival in a country is to make sure that you are admissible and to give you permission to enter if it is needed.
Remember that there is nothing unusual in being a dual national, and you are able to carry and use both the passports that you are entitled to hold. There is no need to show both passports at once, although obviously if someone with reasonable grounds to ask wants you to produce the other passport, you should do so. (It is most unlikely that this will happen, however.) In particular, your being a citizen of both the UK and NZ is not a situation that is likely to surprise anyone who is remotely connected with the travel industry in the UK, NZ or Australia.
The fact that you have never been to New Zealand doesn't matter: New Zealand, in common with Australia and the UK (and many other countries), allows citizenship by descent, and citizens by descent, by definition, may never have been to the country of their citizenship. So there is nothing unusual in the fact that this is your first trip to NZ, and it doesn't make you any less a New Zealand citizen.
As to travel insurance, I would take out ordinary travel insurance for the Australian part of your trip. You are probably entitled to emergency health care, but you should have travel insurance anyway, and the medical coverage will be more comprehensive. For NZ, you need to be sure that you will be covered by the medical system there immediately on your arrival for medical cover; if not, you might want to look at top-up insurance. (Someone else might know the situation for new NZ citizen arrivals in NZ...)
Enjoy the trip and good luck with what sounds like a new life in NZ!
