Insurance - will you buy/drive a car ? will you have personal belongings/valuables with you (camera etc) ? are you covered for health costs ?
Also don't forget that if you don't maintain private health cover in Australia you may need to pay a surcharge if you are still going to be considered an Australian resident for tax purposes. Speaking of which, there may be things you need to do to qualify to be an Australian resident for tax purposes - you'd need to check with the ATO on that (their website should have the info).
Dealing with money in NZ should be pretty easy ... and you can possibly get away without a NZ bank account - depending on how you manage your money.
I suggest you get a couple of Australian credit cards (you probably have these already) and arrange for them to be paid out in full each month from your Australian bank account. Then in NZ, pay for everything you can on credit card, and just take cash out of an ATM as required for day-to-day use. ATMs are common in NZ and pretty much all of them take Australian ATM cards. There are ANZ and Westpac (or the NZ equivalent) banks and ATMs in NZ ... check with your bank, you may be able to get fee-free withdrawals. I suggest multiple cards, just in case you have problems with one while away.
Even if you do have to pay a fee for withdrawing cash, try taking out a larger amount less often - the fee is a usually a fixed amount ($5 typically), so the more you draw out, the lower this fee is as a percentage of your withdrawal.
I recommend you carry an Amex card while travelling - even if you only keep it for emergency use ... Amex have offices pretty much everywhere around the world, and can typically issue you a replacement card within 24-48 hours if you lose your card, or worse - if there are fraudulent charges on your account which require you to cancel your old card. This has happened to me while on a work trip to Singapore for 4 weeks ... Amex were quite helpful and I had a new card the next day.
I wouldn't worry too much about proving you can support yourself ... just make sure you have a couple of credit cards and perhaps $10K or $20K in your bank account. Perhaps check with the NZ Immigration Service beforehand as to what is required before you arrive.
Also remember that NZ has strict quarantine laws - so don't try and take food/plant items with you without checking first. Similarly on your way back - don't try and bring stuff back from NZ into Australia.
As for mail redirection and such - I know there are companies that offer services for people who are travelling around Australia long term (eg motorhome trip around Australia) - they hold your mail and forward it to an address you tell them (eg the caravan park you are staying at, or the post office of the township you intend to be in soon). I'm sure there is someone who can do similar to OS destinations as required.
There are also portfolio-management companies which do most of the work of managing your investments for you (effectively, a personal assistant) ... I don't know how much they cost, or whether they are suitable for your needs - but it's worth looking into this as a possibility for while you are away.
I may be in a similar situation as you sometime in the next few years - we are planning some extended multi-year trips OS, and will need to ensure we have these issues covered as well. I haven't yet started looking into the logistics in detail yet.
Let us know how you go
