National Park railway station: The useful rail option is KiwiRail’s Northern Explorer between Auckland and Wellington, which stops at National Park on scheduled operating days. Book the train rather than assuming it runs daily, then arrange accommodation pickup or walk/taxi into National Park Village; it is a tiny settlement, not a city with an urban transfer network.
Taupō Airport: Rent a car at Taupō if you can: National Park Village is roughly 1 hour 45 minutes away by road, and a car is genuinely useful for trailheads and weather pivots. There is no frequent, turn-up-and-go airport public transport connection worth planning a hiking trip around.
Auckland Airport: A rental car is the practical choice: allow roughly 4½–5 hours to National Park Village, longer in holiday traffic or poor weather. Alternatively take the Northern Explorer from central Auckland on its operating days; do not book an expensive private airport-to-park transfer unless your group makes it worthwhile.
Wellington Airport: Rent a car and allow about 4½ hours to National Park Village. The sensible rail alternative is to get into Wellington city and use the Northern Explorer when it is running; it is scenic but schedule-led, not an airport transfer service.