There are a lot of differences in climate and terrain between Oz and NZ. They are not really comparable (except of course in localized areas). The flora and fauna are quite different, as they have evolved quite separately over geological time. For example, NZ has no snakes, kangaroos (except where introduced), marsupials (I think), and far fewer poisonous spiders (there are some). From what I recall of my schooling, there are no native mammals, other than bats, maybe a native rat or two. Naturally, the beer is superior, also.
In NZ, hammocks ought to work quite well in the bush, providing you are not damaging trees. You are definitely going to want mozzie protection, and a good tarp and parka.
Alpine areas will of course be alpine desert, with no trees. The weather can be quite changeable.
Tongariro National Park (TNP) in the center of the North Island, about 4+ hours drive south of Auckland, is great for alpine trekking and climbing, and skiing in the winter (Whakapapa, Turoa) (wet snow; ice). A volcanic area, it occasionally provides eruptions, and even deadly lahars.
If you fly into Auckland, there are good outdoor shops to guide you (Queen Street (City Center), Broadway, Takapuna; use the yellow pages). The Youth Hostel Association (YHA) is quite a useful network, and there are green buses that run between (some of the) hostels. Great quiet beach hostel: Opoutere. Surf at nearby Whangamata, or further south Mt. Maunganui. Useful mountain hostel: Ohakune (TNP/Turoa).