We took a private tour in Auckland, and had a wonderful day despite it being so windy.
There were also a number of places to visit that I would recommend:
1. Old St Mary's Church in Parnell. Interesting history and it hosts Sir Edmund Hilary's banner from when he was a knight of the garter and the pulpit has a tiny mouse carved into it.
2. Take the ferry to Devonport to explore it. It's really quaint and lovely.
3. Get to the top of One Tree Hill - the view is stunning and well worth it.
We wandered around the city under our own steam, which is possible in Wellington because it's so small.
A tip is to take the cable car to the top and walk downhill through the Botanical Gardens around to the Parliament House (the Beehive) and take a tour there. And don't forget to see the southern hemisphere's oldest and largest wooden building just down the road (now something to do with the law).
The port is also very close to town so it's possible to walk to and from the ship to Wellington.