Still file via NAIPS while in aussie territory so that includes Norfolk however thereafter you will have to go Christchurch Control. The nice people staffing the UNICOM can help you fax a flight plan if you cant get internet. The NZ system is called IFIS,
www.ifis.airways.co.nz.
Radiocalls are fairly similar to australian ones, you will have to use the full Victor Hotel XXX when entering NZ FIR but may be abbreviated to VHX or VXX by each controller.
On the topic of radio requirements -flying in NZ you will encounter areas marked as CFZ and MBZ, that means Common Frequency Zone (much like a CTAF for an area) and Mandatory Broadcast Zones, in the latter you must report entering, leaving and a pos rep every 10 minutes.
VHF Coverage is not great, there is a repeater station for MLCEN on Lord Howe but not on Norfolk. You'll be back in VHF not until visual with NZ I'd say(never tried the distance, went to HF early)
Weather should be stabilizing towards summer, but generally a tailwind going to NZ.
Easiest thing if you want to make sure you dont forget is to visit the customs NZ and customs AUS websites. esp the NZ is very helpful in describing which documents are required. Not many forms, just crew declaration and entry/departure cards.
I have not been ramp-checked by CASA or NZ CAA at all so probably not. Check with the NZ CAA if you need a special flight permit but I really dont think you will. Anyway, the NZ CAA are extremely helpful, much more so than CASA IMHO. Do bring your passport though.
NAV: are you VFR or IFR? -we flew NZ reg planes IFR only. Our RNAV system was certified for oceanic. Providing you fly during the day, you should actually have NDB coverage all the way. Just ensure you crosscheck the requirements with your equipment list.
What sort of plane are you in?
Here's the customs website that lays down the law: in addition you probably need to arrange the ag-people to check for food/pests etc. No cargo manifest unless you are carrying cargo other than personal affects. No cash report unless carrying more than 10k$. Just need entry cards and your clearance from precious port. -this is a list of the crew with the details that will get stamped by the customs when you leave. do NOT lose it.
We never came into NZ so unsure about their procedures.
New Zealand Customs Service : Advance notice of arrival for private aircraft