Hi John,
First off, I hope you enjoy NZ as much as we did. It’s a fantastic place, full of great places and the people are terrific.
As far as flying a drone goes, there are a number of things that cropped up while we were there.
It’s pretty much possible to fly where you want, except that you can’t fly where you really want to. To explain that statement, there are quite a few restrictions around the spectacular and touristy places, which are the areas that you would most like to film or photograph from the air. The Department of Conservation (who do an otherwise wonderful job) are very restrictive about drone flying. You’ll see “No Drones” signs all over the place on land that they control - which is pretty much all the bits you’d want to go to, from Cape Reinga in the north, any of the national parks (Aoraki/Mount Cook or Fiordland for instance) and places like Castle Hill.
There is a scheme for getting permits to fly for some of these places, but after I got knocked back for 3 or 4 applications I gave up 
Added to this, the New Zealanders are (quite rightly) very protective of their marine wildlife, so coastal areas have many restrictions as well, especially around whale or dolphin areas. And there’s bloody birds everywhere!
In addition, there are many small local airfields, as well as the commercial airports, which have their own restrictions. You can work around these to a great extent, but small aircraft and helicopter flying is a problem for drones. I was flying near Franz Josef Glacier, just around the AirBNB we were at and outside of any restrictions, but I had to rapidly land when a low flying egg-whisk made a close pass over us.
Local authorities have their own set of rules and restrictions about flying over their assets, such as buildings, parks and some roads., and the Maori people are entitled to their say about sacred places as well.
Having said all that, there were still a lot of good opportunities to fly. There is an app called AirShare which is excellent, and I suppose it’s down to how much of a rule-follower you are. I tend to be a bit goody-two-shoes when it comes to this sort of thing, so I didn’t fly as much as maybe I could. Still got some good footage though.
I hope this helps you, and doesn’t come across as too depressing. Like I said, NZ is an amazing place - so much so that I sort of forgot about the drone for large periods because of the places and things we were seeing! Hope you have a great time!