From the research I've done (and like PV_Premier said), you really need to drive to be able to do everything at your own pace and schedule.
Why don't you looked into NZ premium economy? I did some random dates in October and November for ORD-AKL then SYD-AKL, and it priced it around $3000.
I'll be leaving soon for a month in for NZ and OZ, so if you have any questions and I remember to return to this thread, I'll try to see if I can help.