An early indication of the public interest in the visit was demonstrated by about 100 people who waited in the airport's arrivals lounge for a brief glimpse of William before he was driven away.
His visit has also generated huge interest among the British media with large numbers following him to the South Pacific nation.
The prince has visited New Zealand twice previously, the most recent time while following the British Lions rugby tour in 2005.
The second-in-line to the throne said this trip was a "bit more official" but he was enjoying "the fact it's reasonably relaxed".
I think it'll be seen as a big moment for New Zealand. And hopefully for the Royal Family
Prime Minister John Key
When asked about standing in for the Queen at the opening of the Supreme Court building, he said: "It means an awful lot because obviously she's extremely busy and I want to be able to do something good and keep the standards up that she's led the way with."
But he would not be drawn on other tours, saying he had to think about his RAF search and rescue training.
"We'll wait and see. I wouldn't get carried away," he said. "I've still got a lot of training to do and I need the training, believe me."
The New Zealand PM told the BBC the prince's visit was a "significant event".
"Having Prince William in his first capacity officiating for the Queen, representing the Queen, makes it even more special," he said.
"So I think it'll be seen as a big moment for New Zealand. And hopefully for the Royal Family."
At the traditional Maori dinner - a hangi - hosted by Governor General Anand Satyanand, the prince lifted parcels of meat from an oven dug into the ground and heated by coals.
Prince William lifts a parcel of food out of the traditional Maori earth oven
And he was welcomed by Maori elders with a hongi - the traditional rubbing of noses symbolising the exchange of breath.
Earlier the prince met New Zealand rugby stars on a visit to the redeveloped Eden Park stadium - one of the venues for the 2011 Rugby World Cup.
William also took to the waters around Auckland when he joined the crew of a yacht used as an early trial boat for the 1999-2000 America's Cup series, won by the Pacific Ocean islands.
On Monday he will attend a barbecue hosted by Mr Key, before travelling on to Australia for a short unofficial visit.
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