The cost of living in New Zealand is higher than USA - our daughter lives in Auckland in Mt Eden in a 1 bedroom flat $320 per week includes water only. Internet & mobile phone costs more than the states, example in the USA we pay a small fee for home Internet and it is unlimited use, mobile phones in the states are unlimited use for a fraction of the cost in New Zealand, these are the two things I miss the most!
Food can be expensive & worth asking people where the local Farmers Market is. ' Pak n Save' is about the cheapest grocery market there are other markets just be aware of price differences. I don't eat out at restaurants much when Im there as it is expensive I cook & bake a lot. I know with kids sometimes you need something quick & Dominos has pizza for about $5.00 which is great, I also like Chineese take away its always a good amount. Save your jars & take away containers, wash them & you will have good use for them.Â
Kmart & Whatehouse have house hold items at low prices - Briscoes has great sales & worth checking out.
I keep a little writing pad with me & I write down prices of items etc... This way I'm able to get items at he lowest price. Bunnings is a hardware store, building, electrical, gardening supplies, etc.. it is worth stopping by here so you're able to know what they have.
Curtains are expensive, I suggest bringing tablecloths or material so you can create your own, I am shipping my daughter tablecloths, ones that are large enough to cover sliding door as well as smaller ones too.
I'm not aware if you're shipping your furniture etc..on a boat or if you're just coming with what you can carry.
If you're coming with what you can carry I suggest that you find out the biggest bag size & weight you can check in on the plane, it is worth paying for extra "Bag's" then paying double the amount when in New Zealand for the same items you left behind.  You can look at the stores I listed above to compare prices now, which I suggest you do to help determine what is costly!
One of the great ways to buy items or find a flat is on TradeMe - I suggest you join this,this is how my daughter found her flat & I was able to buy a fairly new Oil Heater for under $15 - you can buy/sell furniture such as beds, tables, etc, cars, homes, property, rental flats. To join you have to hve New Zealand phone number etc.. I'm sure you can contact them through email check with them about using it now if you need to start looking for a flat. Â
The best way to find out about anything is to ask.
Cheers and good luck to you and your family.