Welcome to New Zealand in advance.
I suggest a little light reading on the subject would be a great place to start. There are several excellent fishing guide books that have been written in New Zealand aimed at the traveling angler both from New Zealand and overseas. North Island Trout Fishing Guide by John Kent is a good one. Another is Volcanic Trout – A Complete Guide to Fishing the Taupo Region by Brendon Mathews, and perhaps 101 Trout Fishing Tips by Ron Giles. This is not an exhaustive list by any means but these would be a great place to get acquainted with the various waters and techniques employed to fish them (links for these books below). Purhaps you can find them in your local library.
You will have to purchase a fishing license to fish in New Zealand from the Fish and Game Council. The license is valid anywhere in New Zealand except the Taupo Fishing District which requires a separate license as this fishery is administered by the Department of Conservation (see link below). These licenses are extraordinarily inexpensive. I suggest you get a family license which is only a few dollars extra. Fish and Game and DOC produce some excellent brochures and pamphlets on the various fishing areas so be sure to ask about these as well. You should also study their websites as they have heaps of information that will help you a lot.
You are talking about an enormous area covering some of the finest trout fishing to be found anywhere. The only places you are likely to find a crowd of anglers are on some of the popular pools on the Tongariro River. Perhaps most famous of these is the Major Jones Pool. You might want to fish it just to say you have been there! There is a certain etiquette to be followed at these busier spots. You can read up about this ahead of time.