Regarding your ‘mission’, you’re in luck. If you’re looking for grandeur and greenery, you’ll get it. If you want to be awe struck, you will be.
Unfortunately, you will miss things and as you only have three weeks it will still seem a bit of a rush as there is sooo much to see.
You’ll be arriving late spring, almost beginning of summer so you’ll be here at the right time for what you’re interested in. Hiring a car is an excellent idea as you’ll get all the freedom and flexibility you need.
From what you’ve told us, and if I were you, I’d spend around a week in the North island and two in the South. You’ll be doing a lot of driving but that’s the way it goes. For the north, the Bay of Islands is spectacular, there’s just something about the place that captures you. The day trip from Paihia to Cape Reinga - the very top of the Northland - via ninety mile beach is excellent and certainly worth considering. The Coromandel is also exceptionally good. On your way south after Waitomo, you’ve really got to choose between travelling through the middle (Rotorua and its mass of thermal activity, Lake Taupo and Tongariro National Park with it’s world famous one day walk) or hitting the East coast (Bay of Plenty, Gisborne, Napier) which has excellent beaches and greenery and also gives you the chance to see some incredible sunrises of which you’ll be the first mainlanders in the world to see.
Though the North island can still adequately cater for some of your ‘would like to’s’, everything else on your list can be easily satisfied in the South Island – mountains, glaciers, awe and grandeur. The Abel Tasman and Marlborough Sounds are relatively close. The beaches of the Abel Tasman are beautiful: white sands and emerald green waters. For your tramping options you could have a bash at the Abel Tasman coastal track
or the Queen Charlotte track around the Marlborough Sounds.
The Tranzalpine is a great experience which will have you back in Christchurch inside a day. Before you get to Christchurch though, Kaikoura (2hrs north) is well worth a visit for the abundance of marine life which inhabits the area. Here you can take a boat and see sperm whales and swim with dusky dolphins in the wild - a most memorable experience.
I would strongly recommend heading down south. The route inland from the east coast through ‘the southern lakes’ (Tekapo, Pukaki (and Mt Cook!), Wanaka, Wakatipu) is an absolute must. Some of the colours are mind-blowing and the vast snow capped peaks of the Southern Alps will leave you speechless. This route will lead you on to Wanaka and Queenstown - both scenically stunning though Queenstown is much more touristy and pricey – but a stunning location nonetheless.
I really think for a once only experience, you must experience the Fiordland – the Milford Sound. Even the Doubtful Sound which is much less touristy and no less dramatic. There are plenty of options including overnight cruises on both, scenic day cruises, even flyovers. You can base yourself in Queenstown, Te Anau or Milford to do these, though the hours drive from Te Anau to Milford is breathtaking. Also, once you’re around these parts you have access to some pretty spectacular treks – the Kepler track
in Te Anau, the Routeburn track
from Glenorchy just to name a couple. You could even coincide your Milford visit with your completion of the Routeburn track as it finishes at ‘the Divide’ - very close to Milford. (The Milford track
is very popular and is often booked up to a year in advance, so be sure to check availability before making definite plans.)
As you’ve already said, the glacier hike (or heli hike) is a must and you won’t be disappointed. In fact, the whole west coast is wild and rugged and is also often missed. You could easily lose a few days there!
I guess it really depends how long you take to absorb all this in the short time you have. Oh and just to throw you right off course, if you can manage to squeeze anymore in, ‘The Catlins’ on the South East coast is well worth a visit, including ancient petrified forests and native seals basking on the beaches.
There is simply far too much to see and do and while it’s unfortunate to miss things, I still reckon it’s important not to rush too much and miss the ‘experience’. If you really embrace what New Zealand has to offer, the quality of your trip will far outweigh the quantity.