Speaking as a kiwi who lives in the US and travels home alot. When you fly into NZ you will fly into Auckland Airport (and you fly out of Auckland as well)so,if you want to see Auckland you can do it at the start or end of your trip.
A great way to travel is the hiring of a little campervan ( I just did it in Nov) check out www.exploremore.co.nz and look at their campers such as the rotovegas. Its cheaper than hotels and rental cars and soooo flexible. You can just drive where-ever you want and pretty much find a river or beach and pull your van over and camp (ok there are some exceptions) but they are very few and far between. They have depots at the airports so its pretty easy… to get from AUckland to anywhere else, your options are by air or by road/rail and it sa great way to see the countryside