We brought everything bar white goods to Australia last year, and we only didn't bring those because I was renting out my house in the UK.
Clothes in Australia are far more expensive than in the UK, I imagine it's similar in NZ.
I did leave some stuff behind as we shipped everything way before we left the UK (my visa took a while) and we needed it while we were in the UK - I am still gutted that I didn't just send my A-frame airer, mind you, as it cost more to freight it afterwards than was reasonable and I can't replace it as they aren't made any more.
We sent most stuff by sea to start with, then some stuff by air (expensive but necessary) and then when I went back in January I did another small sea shipment as some of the bits that I thought I could cope without/buy again hadn't been easy to be without/ replace.
Be careful with buying new goods, especially electrical - I woudl think NZ have the same policy as Australia and they will charge you some kind of tax on new goods. You have to have owned them for a year, apparently, prior to bringing them over here. I don't know how strict they are on this as we didn't try it - all my electrical stuff was ancient.
Definitely bring a UK DVD player if you have lots of DVDs because some won't play on this Region's DVD players - I think we're region 4 and the UK is region 2 but I might have that wrong.
And buy converter plugs in the airport - again, dunno about NZ but they are difficult to find UK>Australia in Australia, and when I do find them they are bulky and expensive. It's cheaper to buy a World Travel converter in the airport than a dedicated UK> Australia one, honestly.
Think twice about bringing outdoor stuff - it has to be meticulously cleaned - any mud/soil/dirt will earn you a cleaning or disposal bill at Customs and it is likely to be more than the cost of replacing the item (unless it's an expensive mountain bike of course).
Make sure you clean all your shoes as well.
BeenBeta - a good friend of mine recently moved from NZ to Australia - she misses NZ enormously, she liked it much better than Australia, she said the people were much more friendly and welcoming. Also, she found that the schools and sports clubs were far more inclusive - very keen on "have a go", whereas in Australia so far she has found that they are more "if you can't do it properly, give up". She liked the community spirit in NZ and has found it lacking in Australia. And she much preferred the weather! But then she is in Townsville, up near Cairns, so that's not surprising.
Re washing machines - again, don't know about NZ but here in Australia they are moving steadily towards using front loaders rather than top loaders because they are generally more water efficient. In Australia, that's a big thing now - we get govt rebates for buying water-efficient appliances. Top loaders tend to damage clothes more as well. We have a Bosch - but I think it was built in Malaysia rather than Germany (the more expensive ones are actually built in Germany). So - you'll still be able to get parts for front loaders here if it's a make that we get here. Check on google.com.nz, pages from NZ only to see which makes are available (you could check Australian pages as well).
shelscrape - if you want your range cooker, bring it with you. I wish I had been able to bring my gas double cooker with me .