Forgive me for writing this at night so I may have forgotten a few good things to try out however there are a whole heap of things to see and do in those places! I’m sure you’ll find something when you are there!
Te Papa Museum is a good time passer. Free of charge and sometimes they have a particular exhibition on for a fee (optional to visit). They currently have Gallipoli: The Scale of Our War exhibition (free) on which I recommend if you’re into history.
Weta Workshop have a studio tour showcasing some of the things they do. You may have heard of them, they worked on Lord of the Rings/The Hobbit, King Kong and many other films.
Mt Victoria gives good views if you’re keen for a walk.
If you’re into architecture/buildings, there are Art Deco tours available (or you can go look at the buildings for free by yourself).
There are also good vineyards in Napier/surrounding area.
You can hire a bike and ride along the path along the shoreline, across a couple of bridges.
The Bluff Hill lookout gives a good view of the port and ocean so a good photo op here.
Don’t go for a swim in the ocean there as there is a very strong undertoe that will sweep you out. Signs are in the informing the no swimming area.
This would be the North Islands tourist city. There’s quite a bit to do here. The Luge is the first thing that comes to mind where you go up a Gondola and ride down the hill on a kart (luge).
There are a few lakes in the vicinity which provide good photo ops and walks.
I recently heard of the redwood treewalk where you walk across suspended bridges spanning through the trees which I personally want to try out.
Zorbing is another activity in Rotorua you can do. It was invented in New Zealand so give that one a go!
Hamurana Springs is a great little walk (20min) which takes you around some redwoods near a stream that is crystal clear. The water from this stream comes up 75m below
New Zealand’s tallest structure, the skytower has a great viewing platform at the top. If you’re not too afraid of heights, you can jump off of walk around the tower at the top. Speaking of heights, you can also walk across the top of the Auckland Harbour Bridge or bungy jump of that too.
Near the Skytower is the SkyScreamer where you are strapped into a slingshot like ride and shoots you into the air. I loved the thrill of this and through it was definitely worth what I paid (I can’t actually remember what that was now though).
If you’re a lord of the rings fan, why not give the Tongariro Crossing a try! It’s one of New Zealands greatest one day walks. I did this last year and it took 7 hours including stops and lunch at the blue lake, however the views were amazing. There are also other shorter great walks in the area such as the Taranaki Falls (2hr loop)
Forgotten World Adventures takes some golf carts and puts them on a decommissioned rail line. Go through tunnels and over bridges - I have only heard good things about this.
New Zealand’s largest lake. Go take a bungy. Hitch a ride on the Huka Falls jet boat which gets inches to cliff faces as you ride past. The Huka Falls is also a great place to visit. You can also skydive in Taupo which has one of the best views when falling.
Want to get up close with an actual Kiwi bird, visit the Ootorohanga Kiwi House.
Waitomo Caves is a very touristy attraction and has many visitors daily. I recommend the glow worm caves experience (no flash photography!)
One thing I think is a definitely must do, is fish and chips on the beach. and if you can meet local people, ask about a Sunday roast and see if you can try this, even if it isn’t on a Sunday!
As some of these things are marketed to tourists, prices might be quite costly for what they are however they may be worh it!
If you have any other questions, I’ll try ask them as best I can.