Spend some time down in Waitomo, there's some amazing caving trips down in that area. Can easily spend most of a day exploring the underground.
You sticking mostly to North Island, or going to spend some time in the south as well?
I miss home, and I don't live there now, but I can at least point out some fun stuff:
AdventuresWaitomo Caves
Surfing, usually southwest of Auckland.
Lots of beautiful beaches and opportunities for ocean kayaking to the southeast, and decent stuff due north. But I think its mostly better in the Waihau bay direction myself - lots of beautiful carved rock and whatnot.
To the Northwest of the cities there's a fun place called tree adventures, I think - like an obstacle course, but well over a dozen feet above the ground.
Jumping off the Auckland bridge, the skytower, or hitting that... sling thing I can never remember the name of, right in front of the skytower. All are fun.
Also plenty of places to go hiking - I really like over near Ruapehu, because you can go skiing if conditions are alright, and swimming and hiking if it isn't right for swimming. There's a really cool trail with a big ass waterfall and a different, smaller waterfall you can jump next to and follow it into the pool. Its a reasonable height and not as terrifying as some of the similar jumps they have around. If you're interested I could probably be more specific.
FoodEnjoy it while your in Auckland because it will get harder to find the further away you get. There's good food around elsewhere (really nice sandwhich shop southwest of Dunedin, great place) but Auckland has the best quality density and variety. Wellington and Christchurch isn't bad either. Soooo...
Get some Turkish pizza on main street. That stuff is fucking delicious. I will be envious.
Angie's Kitchen, on Mount street (just up the hill, off wakefield, off Queens) - delicious Mamak Me Goreng. Amazing. 99% of the time. If they try to give you shit with frozen vegetables, send it back and tell them to give you some real food damn it.
Get some Hell Pizza or some Murderburger, if you're in the mood for pizza or meat. Hell also does dessert pizzas! (I don't actually think Hell is that great, but the desert pizzas are something I haven't seen elsewhere.)
Around AucklandKarangahape Road is good for nightlife - good bars, concerts, that sort of thing.
Lots of nice parks, and good museums. I like their... tech museum, I forget what its called. Auckland art museum exists if your into that, I guess... If you're interested in Museums, I'd head down to Wellington instead.
If you go to south Island:Spring&Fern bar in Nelson, great beer, let you fill your own bottles. I ended up filling three milk jugs with the "Scrumpy" wine to take with me last time I was there.
Ride the little train around when you get off the ferry! It's only a quarter, and worth it for the picture and because the guy who runs it is awesome.
I spent a week hiking in Kahurangi park - spend at least a day or two there yourself if you can. It's some of the most beautiful hiking scenery I've seen. Even the giant carnivorous land snails were beautiful!
Te Anau is a lovely little town with same great and easily explorable wilderness just over the lake (walk around or take a boat). I've spent maybe two weeks there, total, and I've never not seen a rainbow, and usually it's several per day, just hanging out.
Fiordland is absolutely beautiful, and Keas are awesome! In fact, this might well be the most beautiful place on EARTH I've ever seen - there's one part with a long tunnel that comes out onto this lush valley, this huge gentle slop down the ocean, the road curving away, birds everywhere, cliffs to either side with dozens of waterfalls racing down them, and the whole thing covered by this glistening fog. It helps if the people your with, like the ones I was, have prepared the themetrack to Jurassic Park and play it just as you come out of the tunnel into the sunlight.