Hi,
In your position I would stay at a motel in Auckland (there are plenty near the airport), hire a car and look around Auckland for a few days. It is a city and, unless you are confident behind a campervan wheel, a hire car would make getting around easier, especially with jetlag. (That's just my personal choice.) Take the kids up the Sky Tower
SKYCITY- AucklandSKYCITY- Sky Tower, Kelly Tarlton's
Kelly Tarlton's Antarctic Encounter & Underwater World, maybe to Auckland Zoo
Auckland Zoo Homepage which has a tram to the neighbouring Museum of Transport and Technology
Explore MOTAT, Auckland's Museum of Transport and Technology. If the kids then behave themselves whilst you are looking around(!), there is Rainbow's End
Rainbow's End N.Z.'s Premier Theme Park.
In Wellington, you could try for a bit of education about the country you plan to move to
http://www.tepapa.govt.nz/Tepapa/English/ (which is made fun for kids). In Christchurch they are sure to like the Christchurch Gondola
Christchurch Gondola - Christchurch Attractions, Christchurch Restaurant & Christchurch Activities.
Haven't used a campervan myself, but looked into it. As March has the Easter break, I would advise having a daily itinerary and booking the ferry from Wellington to Picton in advance, as I hear it can get busy, especially for campervans. You should certainly be able to visit and drive between these main centres in three weeks.
