Suggestions:
Rotorua - to see geothermal stuff. If you talk nicely to the locals they may tell you where the free natural hot springs are (they are on just off the Rotorua to Taupo Road, after the "Murupara" turnoff). The Polynesian Pools (hot springs that you pay to get into!) in Rotorua are good too.
Central Plateau moutains - in the middle of the north island are quite scenic (and on the main road). There are some nice one-two hour walks there.
Wellington - centre of government, does have the national museum, which is reasonably good (and free).
Nelson - Abel Tasman kayaking - beautiful area, and can do guided day trips that take you into the national park
West Coast South Island- a wallk up the Glaciers is a must.
Queenstown - adrenalin junkies, but is quite pricey (but hey, you're on holiday)
Christchurch - if you can get there on the weekend check out the Arts Centre markets. Lots of locally made arts and crafts.
Kaikoura - whalewatching is another biggie, although be warned that this needs to be booked in advance!
Back to Auckland quickly!?! (if you are flying out of Christchurch it is easier!).
The Coromandel is also nice - you can get a taste of it by stopping at Hahei beach - there are nice beaches (including a hot water one - more hot springs), and a good backpackers there (Tatahi lodge).
Also, Waitomo to go black water rafting (not as scary as it sounds - basically float through glow worm caves in an inner tube)is also worth it
All in all, you will be pushed for time, but with careful planning it is do-able.
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