Not an NZ person but when I was there (note, this was > 15 years ago) we took the ferry to Devonport which is an island across the harbor. Some wartime ruins and museum-y things. Speaking of which the Auckland museum is great, had surprisingly good food and a cool earthquake simulator. Also the Maritime Museum, if you're into that (which I am). Waitakere park (not sure if it's public-transportation accessible) has a beautiful waterfall.

Not sure how tour-able these are from Auckland, but: hiking Mount Doom (Tongariro) is cool; Rotorua is also really interesting, volcanic hot pools and Maori culture.