Close to Auckland - The Coromandel, areas around Warkworth, Devonport, Waiheke (definitely), Great Barrier (if you want an adventure). South Island - Queenstown (really touristy, but so lovely), Wanaka. A trip on the Earnslaw is great and Glenorchy at the far end of Lake Wakatipu is a magical place. You can get the Dart River Safari from there - expensive but really worthwhile. Arrowtown is very interesting and there is a collection of old miners houses on the banks of the river which are very moving.
Dundin is a very young and funky little place with a good arts scene.
Further north, Nelson and Blenheim are very foody and there is a cycleway around Nelson called the Taste Trail that looks great. Heaps of wineries around Blenheim. Close to Nelson are some great beaches - try Rabbit Island and Tahuna for relaxed days out. Kaiteriteri is also a really popular beach destination and Marahau is also lovely. A little further afield and over the Takaha Hill is the little alternative town of Takaha. Very laid back and at the centre of Golden Bay, which is famous for its wonderful golden beaches. In this area you can do a safari to Farewell Spit, go fishing at Anatoki Salmon and visit the dunes at the wonderful Wharariki Beach. Between Golden Bay and Motueka is the Abel Tasman National Park - there is a Great Walk that runs all along the coast (takes 4 days), but you can do parts of it and get in and out of the park by water taxi. Also lots of sea kayaking opportunities.
Kaikoura on the east coast is famous for whale watching, swimming with dolphins, dramatic coastal walks and its Maori and whaling history. The town is a bit rough and ready round the edges, but very interesting.
The west coast is much more isolated and wild - 'Coasters' are famous for being staunch and rugged.Karamea really is the back of beyond if you like that whole middle of nowhere' really beautiful thing. Further south is Punakaiki or Pancake Rocks and this whole area is wild and stunning. The glaciers are south of pancake rocks and are worth a visit - i like Franz Josef the best and you can visit Shantytown on the way - old historic tourist attraction in a people dressed up style. Hokitika is famous for its greenstone and lots of carvers work from here. If you keep driving down the West Coat you eventually get to Haast (nothing there, really isolated) and from there you can drive over the Haast Pass and eventually you end up in Wanaka.