I've just emigrated to NZ from London and flew over with my 18m old son less than a month ago.
1st thing I'd say is that flights via the US have double luggage allowance which for us meant a lot more stuff that didn't need to be shipped and that we'd have with us immediately.
We flew with Air NZ, they were good. Previously (when childless) have flown with Air Korea and JAL. Air NZ beat them hands down.
The first leg of our flight was not fun (London-LA). Second leg (LA-Auckland) pretty good and final bit (Auckland-Wellington) ridiculously easy. In retrospect
I think a lot of the 'problem' was my husband and I trying to make our son 'behave'. By that I mean we spent a lot of the first flight trying to make him sit in his seat (we paid for him to have his own seat). In contrast on the second flight my son stood in the aisle for a lot of the flight and was so much happier. Basically we were trying to force him to do things our way rather than adapting to make it possible to do things that he wamted to do. I'm not sure I wrote that v well, but hope you get my gist.
We took our carseat on board and for the first flight used it on the plane. Lots of good reasons for this, (safe, he was used to the seat, more comfortable for him to sleep), but in retrospect again, it was probably a mistake. I definitely recommend taking the carseat tho, v convenient for taxis to/from airport, hirecars, etc.
Erm, what else? Sticker books! Books/toys that make noises (plane noise cancels out a lot). Bizarrely, when nothing else would work my son was happy to just stand in the bathrooms, maybe because the light was bright, who knows? But worth a try.
Oh yes and chocolate buttons!!!