Routes to NZ, Oz and S America do indeed get very busy at Xmas - if you leave it too late, you may not get a seat, and they're unlikely to get cheaper unless there's another war or health scare (god forbid).
Further advice depends on how you're planning to travel - by Air NZ (my favourite, but not the cheapest) or cheapest available (could be via Japan or Indonesia) - stopovers etc. Have you looked into Round The World tickets? There's often good deals to be had on these.
However you get there - it is the best place for a holiday in the world (just MHO).
