As a United States citizen, you only need a valid passport to visit New Zealand. If you stay longer than three months and less than nine months, you will need a visitor's visa. In addition to your passport, you will need travel tickets or confirmation of your planned itinerary and evidence of your ability to support yourself while you are there.
Rent a car and take one of the ten themed highway tours that include the treks along the coast, through historic towns, across inland routes, and through wine growing areas (See Resources). Another option is to see the country on a tour, whether guided or independent.
Whether you stay on the south or north island, choose accommodations from the luxurious to the most basic, from boutique inns to backpacking locations. Choose according to location, price, and length of stay.