I'm actually in Auckland at the moment, living here til April (that's the plan so far), so I can tell you a few things to do.
First thing I'd do in Auckland is not bother with the city.. Its nice, but its a city. Luckily from the harbour (edge of the city centre) you can get to wahieke island, its a half an hour ferry from the city, (and my missus works on the ferry, so say hi to fiona). Its a beautiful island with vineyards and beavhs and tiby 'settlements.. Just how i expected rural new Zealand to be like. If you can spend a night there, feel free. Though you can have a lovely day out.
Anyway, another place to see is piha beach, its a black sand beach near Auckland.. Something rare to see..
Also on that coast is the glow worm caves if that is your sort of thing.
Other places that sound interesting are Taupo and Rotorua, wangarei in the north is also a good place to go.
While in Auckland (another thing to do) you can look at rangatoto. Another ferry trip, but its a baby volcano with black sand/rock/surface. You can go for a good 2 hour hike.. Apparently its amazing.
If you have any questions about Auckland, ask away.. I'll check back often.