I too married a New Zealander and I had a three year old to ferry back and forth. My son and I spent a lot of time together on airplanes across the Pacific ocean. Here are some things I found helpful:

Before the trip, make sure to involve your child in the preparations. Get some books from the library on airplanes, flying and also your destination. Find NZ on a map. Let them help with the packing. The more knowledge a child has about a new situation, the less likely that he/she will be fearful/over-excited. Leave yourself plenty of time to get to the airport. Stressed parents make for stressed kids. Make sure that they see you are happy to be flying, and excited about the prospect of travel. It should be talked about as a grand adventure, not a grim trial to be survived. (Even if you think that it is, tee-hee!)

This stuff may be obvious to you: bring the foods your child likes to eat, because the airplane food (even the kid’s meals) will be nasty. Bring a comfy pillow and his/her favorite stuffed animal. Also a warmer blanket. Lots of not-too-sugary, no caffeine drinks. Puzzles or legos are murder on a plane because they will be forever dropping the pieces and you will have to get them. But those painting books with the dots of paint already printed on the paper are really good. You only need water and a q-tip for entertainment. Ditto coloring books, picture books, and sticker books. Think about getting (or making) some children’s stories on tape or CD for them to listen too. I found with my son that movies were stimulating, and might keep him awake, but listening to bedtime stories or kid’s tunes would put him to sleep.
On stopovers try to find the kids area with a playground. Most airports seem to have them these days. Let your child run around like a mad thing for an hour or two, while you sit and sip some tea. Then go shopping for small things at an airport kiosk or shop. For some reason kids find this absolutely great fun.
Above all, pack a lot of patience. I usually had less trouble tolerating my kid than I did other people on the plane who obviously have never had any children, and don’t like them. Although most passengers are quite nice, you will inevitably run into a grumpy one now and again who turns around and snarls if your child kicks their seat, or makes too much noise. Give them a smile and a shrug, but don’t go overboard with the apologies. Kids have a right to be kids, even on an airplane, and trying to stifle them completely just adds to your stress and theirs.

It is tough, the first time, but tell yourself what a wonderful time you will have when you arrive here in NZ. Good luck!