You can get to the central city by taxi, airport bus or a fixed-price 'shuttle' van that will drop you off at your destination. Taxis have reliable meters that calculate the fares - bargaining and tipping are not practised. The taxi driver can estimate the likely cost for you. Airport buses and shuttle vans (door-to-door) are significantly cheaper and just as reliable.
All cities have bus and taxi services. Only Wellington and Auckland have commuter train systems; there are no underground rail networks.