I've done the shift myself, my recommendations:
- Get an AU number as soon as you arrive, use that as your daily driver... even for calling NZ/SMS/MMS.
- Shift to prepaid on 2D or One NZ. Either one will allow you to use Wifi calling and set up voicemail. 2D will allow you to forward calls (at a cost).
- I don't recommend carrying a 2D sim with large balance roaming a lot as they do have the $8 a day roaming and that can eat into that (I've accidentally burnt a $20 top up through that). I normally only turn on my NZ number once or twice a week tops, or as needed.
- NZ Banks will accept AU numbers, you just need to call them. I've only found Milford to have issues (using email as 2FA instead).
Before someone asks, I can't get an Australian card yet as I have liquidated my assets and investments so I have no income. I need to show some income. That will happen in about a years time.
Many banks will allow you to open accounts through the use of a drivers license + medicare card and don't require income tests. Suggest looking at ubank or ANZ Plus (both digital interactions).
Both of these will allow you to get Apple Pay/Android pay straight away without needing to wait for the card.