North of Auckland is really nice, Warkworth etc. Whangerei isn't up to much keep going and head up to Russell, Keri-Keri, the Treaty House at Waitangi if history interests you. As a kid I loved the old wooden churches with the bullet holes from the wars (I can't remember what I have to call them now as a kid we called them the Maori Wars).
Second the suggestion of Waitomo Caves with the glowworms.
Rotorua also worth a look, and just generally try and go on a Marae visit if you can find one that isn't too tacky, make sure somebody takes you to a Hangi at some point. And just take a visit to South Auckland, Otara etc at some point, the markets etc for a day out.
If you like beaches and surfing Raglan on the west of Auckland (but watch out for the Bogans).
Napier has loads of stunning Art Deco architecture if thats your cup of tea.
Martinborough over in the Wairarapa north of Wellington, fantastic vineyards I often spend a day with my folks on the wine trail there when I visit.
New Zealand is often billed to tourists as the land of adventure, jumping off bridges etc but there is so much more, a rich and diverse culture. You can jump off a bridge in your own country...