Bob,
As several people have said, go to Classic Flyers in Mt. Maunganui. It’s a must. A great museum with a very nice attached cafe. Quite a few of their aircraft are operational and some are available for joyrides, usually a Stearman (1 passenger) and an Ag Cat (2 passengers). If you ask nicely, they might show you the amazing aircraft in the NZ Historic Aircraft Trust’s hangar.
https://www.classicflyersnz.com/Flights.html
In Auckland, MOTAT and the War Memorial Museum a must. Their Spitfire XVI is somewhat depressing to look at in my opinion - marooned forever on an upper floor with only 638 hours in the logbooks!
https://www.motat.org.nz
Home - Auckland War Memorial Museum
South of Auckland at Ardmore, is the NZ Warbirds Association hangar, which you can visit. Lots of mostly operational aircraft
https://nzwarbirds.org.nz/
If you are tired of aircraft museums, I can recommend Hobbiton, which is easily reachable by car from Mt. Maunganui
https://www.hobbitontours.com/en/
and also the Glow Worm caves at Waitomo. You can do sedate or adventurous trips:
https://www.waitomo.com/
I did the Black Water Rafting Black Abyss - an amazing adventure:
https://www.blackwaterraftingwaitomo.co.nz/5-hour-tour