In my experience, some of the best bits of NZ are near the tips. Cape Reinga, Bay of Islands in the north, Waikaremoana, East Cape in the east. Fiordland, Te Anau, Milford Sound, etc in the south.
I haven’t been south of Christchurch in tourist mode, so my experience runs only as far south as Akaroa. It is beautiful, and has the added attraction of viewing or swimming with the Hector’s dolphins, an indigenous species that’s about 1.5 m in length.
The Tranz Alpine train is great. We did a return trip in one day which was a bit of an overload of West Coast forests and alpine scenery. As Greymouth does have a beautiful beach (not one I would swim from though) and other attractions, take a one way trip and pick up a hire car there to travel the very uncrowded roads of the west coast. Go south to Fox Glacier, where you can explore around the foot of the glacier which comes to within 250m of sea level, right in the forest. After, head north through Greymouth, home of Monteith’s Brewery. They make a great tasting range of beers still, even after being taken over by NZ’s biggest brewery, Lion Nathan.
Stay at Westport overnight so you can take a day trip to Denniston. My grandmother was born there, on a plateau 1700 feet above sea level. The Denniston Incline once lowered full railway wagons of coal down what is up to a 1:1.25 grade. Even if railways leave you cold, this is a great spot to view the West Coast (provided it isn’t raining).
The top of the South Island has great beaches in Golden Bay, great scenery in Abel Tasman National Park and lovely small towns right through Tasman Bay to Nelson. Next door to Nelson, in Marlborough, there are extensive vineyards and wineries. There are also vineyards in the Wairarapa in the North Island and around Napier and Gisborne on the east coast.
Wellington, the capital, is not bad (I say that as a long time resident in Hamilton, one of the boring cities (go on, look it up)). The capital has Te Papa, the national museum, and there is a lot there for such a young country. (Only just 200 years of European occupation.)
Unless you have a fascination for cities, I’d not bother too much with Auckland. It used to have the moniker City of Sails, and it has a huge number of yachts that take to the harbours and Hauraki Gulf at weekends. It also has big traffic problems, something perhaps due to the roading being built 5 years after it’s needed. Most tourists just treat it as the gateway to New Zealand as it does have the most connections to other places, internally and externally.
(Added 11:57) Okay, most of what I wrote would be for a freedom traveller. Sorry. Still, Milford Sound and Cape Reinga are both places that have regular tourist coach trips to them.