There's no train connection between the airport and Wellington.
Car and bus
Wellington International Airport is situated eight kilometers southeast of the city center and is easily accessible by rental car. The airport lies at the end of Highway 1. From Wellington's center you therefore simply drive to Highway 1 (well signposted everywhere) and follow the signs to Rongotai/Airport. You drive via the Mount Victoria Tunnel.
Every 15 minutes the Airport Flyer runs from the airport via the center of Wellington and Queensgate to Hutt Valley, with Upper Hutt being the final destination. This bus service offers free wireless internet on board!
Car rental Wellington International Airport
Five car rental companies are present at Auckland Airport: Avis, Budget, Hertz, Europcar and Thrifty. The counters of the rental companies can be found in the Arrivals Hall. You can also book a rental car in advance using the service provided by EasyTerra; they compare the rates of several providers in a single quote.
Taxis can be found outside the Arrivals Hall. One line is reserved for cars of Wellington Combined Taxis (Tel. +64(0)4 384 4444), the other line is for all other taxi companies.
The short-term parking places can be found just outside the terminal. At the uncovered parking lot you pay NZ$ 3.50 per half hour, in the garage NZ$ 5.00 per half hour. The long-term parking lot is further away, you pay NZ$ 27.00 for the first day, and then NZ$ 12.00 per day. Shuttle busses maintain the connection with the terminal.