As an Aussie, I can't give you a NZ-specific answer - although given the similarities between Aus and NZ, I'd be surprised if there was a huge difference.. The WPATH Standards of Care will also play a part..
The basics to getting started are fairly straightforward here - I went and saw my GP and requested a referral to a specialist Gender Therapist, I had all the therapists info when I went in to see my GP. I then saw my therapist for a while and he then referred me to another specialist to prescribe and monitor my HRT..
Now, I haven't mentioned hair removal.. or voice training.. or facial surgeries.. or SRS - but I have given you the basics of how to get started on HRT..