I agree with the others who have suggested that, if you don't want to rent a car and want to stay in the city or cities you fly into or out of, Sydney and Melbourne are much better choices than Auckland (or Wellington, for that matter). The most interesting and beautiful parts of New Zealand are outside its two largest cities. That said, it is easier than you might think to get around New Zealand without a car if you want to. I had a great time hopping from place to place in a 10-day period via plane and train a few years ago. One-way flights on Air New Zealand were relatively cheap, even at fairly short notice.

One possible itinerary could be: (1) Auckland to Wellington by plane (and explore Wellington for a day or two); (2) then take the ferry to the south island and hop on the train that runs along the coast and then down to Christchurch. You can stop along the way in some interesting, small coastal towns (e.g. Kaikoura for whale-watching). From Christchurch you can take the TranzAlpine train across the island to Greymouth and back again. Then fly Christchurch to Sydney. That sounds like a lot of moving around in a short period, but it's not nearly as hectic as it sounds.