http://wikitravel.org...riving_in_New_Zealand - note the quirky left-turn rule.
As long as you're going outside school hols you shouldn't need to book rooms except for the beginning and end, just go where your fancy takes you. Camper vans are a popular and well accepted way of doing it. Towns (and motels) are quite widely spaced, especially in the South Island - buy motel guides from the AA. Motel usually have kitchenettes attached so you can cook for yourself.
Three weeks the minimum, it's a long way. Jan - April is the best weather. (I'm going there in Feb).
What sort of thing to see? Thermal area in Rotorua, Skytower view in Auckland, Weta Digital minimuseum in Wellington, whales in Christchurch, Tranzrail trip Christchurch-Greymouth and back, bungee jumping in Queenstown, etc etc - maybe even a rugby match.