Hello Eric - I lived North Island for many years, but am not as familiar with the South Island so can't advise you much on that.
Regardless of which island, be prepared for changeable weather. November in NZ is comparable to April or May up here. The Rotorua and Taupo areas are your best bet on the North Island at that time. A great guide in that area - who also has a Bed & Breakfast right at the mouth of the Ngongotaha - is Lyndsay Lyons.
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As with many places, a guide on day one can yield some good self-guided fishing on subsequent days. Lyndsay, or any of the other guides in the area, can guide you into some less obvious water and bigger fish.
Not all NZ fishing is long leader sight casting to big spooky browns in gin clear water! There is also a lot of streamer fishing, nymph fishing, and pocket water fishing with dries. I'd suggest a 5 weight, but nothing smaller. Take the usual leaders and tippet you'd use on larger trout over here. For sure take a box of nymphs, but many of the streamers used there are quite different. Most streamer fishing in the North Island is imitating local chubs; my NZ streamers are unlike anything I use over here.
And most important - take lots of sun protection. That hole in the ozone layer is a real issue down there.
Tight lines,
Ric