Hi, we flew long haul for the first time when we emigrated here, had a 1 year old, 3 year old and me 27 weeks pregnant (yes I was mad! 
), so I think with the age of your DC you should be fine 
.
We flew Air NZ, which we found really good, friendly and great with our kids. We went with the sky couches as we had such little ones, which turned into a sort of bed/playarea which worked well for us, but with the age of your DC probably isn't necessary, plus if the people in front recline their seats you get squashed! I've also heard from friends and relatives that have done the flight that Singapore Airlines are brilliant.
We went all the way in one go, with just a 2 hour stop off in LA as we just wanted to get it over and done with but if they'd been older I would have wanted to stop somewhere on the way. My eldest DC is 5 now and I think she would enjoy stopping and seeing somewhere. Singapore is a popular stop off, the airport there is supposed to be fantastic, with a swimming pool, play areas etc, and my mum did a tour of the city whilst she was there. I know Hong Kong is another popular stop off, and there is a Disneyland there apparently...
As for the flight itself, the TV's on each seat show loads of movies, kids programmes, have games etc on them, which should keep them entertained for a good while. We took a portable DVD player on too with some of the kids' favourite shows in case they didn't watch the airplane TVs (which they didn't...) and lots of books, stickers, colouring etc. My DD liked walking up and down to get the free snacks!
Whereabouts in New Zealand are you headed? It should still be pretty nice weather here in April :)