Ok, food and culture--Rotorua has opportunities to see the local Maaori culture and try hangi (food cooked in a traditional way) so that might be a good option for you. There are some nice B&Bs and local food options north of Auckland, which might be an interesting thing to try as well. If you want to "stay with a local", some places offer "farm stay"--being local myself I've never used one, so can't vouch for quality, but I'm sure some internet searching could help! Te Papa museum in Wellington is a must-do, and if you are going to be there during the World of Wearable Arts that is pretty cool too.

In Auckland the musem gets good reviews, and you can eat whatever takes your fancy--might be worth building your tour around that. As an aside, unless you are picking your car up that night, if you plan to stay in the central city book a shuttle rather than a taxi--taxis are very expensive and the shuttle is quick and cheap.