You could probably do the “major sights” of both islands in a week, but it will be rushed.
We hired a car – between 4 people, the cost is pretty good, and it’s the best way to get around really.
Arriving at night, we spent the next day in Auckland, then headed up to the Bay of Islands the next morning. 3 days/2 nights up there, though 2 days might have been enough. Then a day driving down to Rotorua, stopping along the way heaps, for all the great scenery. After spending some time in the hot springs, a half-day drive to Napier (try to time it so you arrive before it gets dark; the roads are tight and not easy). Overnight in Napier, then half day or so to Wellington. 2 days in Wellington before flying back home. Total 9 days, a sampler pack of New Zealand more than anything :)
It all depends on how willing you are to drive, really. There’s an easier route between Auckland & Wellington, but we criss-crossed the north island so we’d see more of it I guess. Bay of Islands is must-see, as is Lake Taupo, just for their sheer natural beauty. Napier is skippable; we used it mainly as an overnight stop. Not that it isn’t beautiful! :)
You could also spend more time in Auckland, but it is just another city really.
Good luck, and I hope that helped a little!