Domestic flights in New Zealand are habitually cheaper than driving or taking the train, particularly if crossing between the North and South Islands is necessary. Airlines function on electronic ticket system. You can book on-line, by telephone, or through a travel agent. You can bring your own bike, as well as rent a bike in some of the larger cities if you want to hit the road. You must carry a helmet while riding, otherwise you may be penalized. Also bear in mind to ride on the left. You cannot traverse on motorways in New Zealand - just be conscious that the only bridge over the Auckland Harbor is a motorway, so you'll have to take a ferryboat or cycle around the port.
Buses are a comparatively cheap and environmentally friendly way to get around New Zealand, though note that services even between main towns are generally only once per day. Most roads in New Zealand are fairly slender and windy, and travelling a long distance in a bus can be a safe and calming way to voyage. Driving around both the major islands by car is usually not a problem. You can reach roughly anywhere you might need to in a two-wheel-drive car or even a small camper van. The amount of traffic is usually low and drivers are generally chivalrous.