My wife and I recently returned from a summer (their summer) trip to NZ. Coming from the USA, and being only advanced-beginner trampers (though with a good level of fitness), we stuck with the "Great Walks" and did 5 of them.
Above all else, my advice would be to start out with what Kiwis call an "easy" tramp and see how that feels. Other Americans I met generally agreed that the Dept. of Conservation's idea of "easy" was often our idea of moderate and "moderate" was rather exhausting. Also, when the rain, cold, wind, or mist is on -- and it will be -- you may find those "easy" and "moderate" hikes go up a notch on the scale, since it can be exhausting to walk all day through heavy weather.
Speaking of rain...we spent a fair amount of time in Fiordland. There, we did something like what you describe yourselves contemplating, Alex. We stayed in Te Anau, went out for a few days, came back, did laundry, bought food, rested, and then went back out; rinse, repeat. Fiordland is beautiful -- and, by the way, you can drink almost all the water w/o treatment -- but if you do any tramps down there, don't merely go with "good" rain gear; take the best rain gear you can buy, since the amount of rain can defy believability. (On the Milford Track we got about 40 cm of rain over 4 days with two days/nights solid of fat drops.)
Also, it is not unusual to get temps near and below freezing along with high winds and rain or snow, generating extreme wind chill conditions with wet clothes. (We started the "moderate" Routeburn Track on a day with 10 cm of rain, 70 km/hr winds in the valleys (the gusts were still higher), and temps. of about 4 deg. centigrade. The next morning, the rain continued and we hit snow a few hundred meters up with continued high winds.)
We did the Routeburn in a tent and the rest in huts. Advantages to both (tent: solitude; huts: warmth), but due to the weather, I'd suggest huts when available.
Lastly, consider buying merino wool layers (e.g. Ice Breaker). We ended up replacing ALL our synthetic/poly- clothes, which simply couldn't handle the huge range of Fiordland weather conditions combined with "sweating when heading up and freezing once you get to the top." Our summation was that the Kiwis figured out something with this merino wool, and life certainly got easier once we switched.
Lastly, forget about this backpacking light stuff when it comes to meals. The Kiwis eat well on the trail. I thought we were clever with our little titanium cups and pots and freeze-dried cuisine until we were forced to watch Kiwis in the huts eat fresh salads, soup in soup bowls, burritos with fresh avocados, cutting up veggies for cous cous, with wine, etc. Unbelievable. Actually, they seem to be a bit competitive about cooking; kind of like the aluminum version of the iron-chef. Our last three trips we packed fresh food, since, really, the Great Walks are relatively short.
My hats off to all of them for their apparently endless enthusiasm and energy for tramping, tough weather and heavy packs, and my thanks for learning how to really eat on the trail.