Hi AmandaB, Havent been to the north island yet but traveled the south and I would say to best satisfy your experience and expectations of New Zealand your family should rent a large
Britz Campervan Hire & Car Rentals New Zealand - Motorhomes, RVs and Cars to Rent ( one of many rentals to choose from) and stay at the holiday parks, most had a kitchen, showers, and TV....I would guess a car would be uncomfortable, for the 3 kids plus It would get the family into the grocery stores to see what foods are different and same...(try the hokey pokey ice cream)... and if capable, rent some bikes... the pavement is smooth,,,, on the other hand if the Bay of Plenty will be your homebase for exploration a rental may be an option....other than that good luck and have a great research visit. Duke
