Didn't mean to post yet!
In Auckland - of the top of my head -
Agree with Justabout, Waitakere ranges. There are some train rides through it some weekends in summer www.waitakeretramline.org.nz/home.html and www.watercare.co.nz/community/rain-forest-express/Pages/default.aspx
West coast beaches (Piha, Karekare etc)
Rangitoto - ferry from the CBD then train or climb to the top, great views of Auckland.
Zoo, plus Western Springs park right next to it, with lake to walk round and a good playground
MOTAT - museum of Technology and Transport - some of it a bit random but quite fun
Kelly Tarltons underwater world place, haven't been for years, not sure how good it is.
North Head, Devonport - old military tunnels to explore, Devonport suburb has nice cafes, bookshops, ice cream shops, waterfront playground.
Lovely beaches on the North Shore, loads of em!
Wynyard quarter at the harbourfront in the CBd, has a tram and often kids activities weekend/summer. Maritime museum nearby.
"City" beaches at Mission Bay, St Heliers, Kohimarama, nice cares, ice cream, playgrounds etc. Near Kelly Tarltons.
Ambury Fam Park in Mangere Bridge, regional park with a working farm on, can watch cows being milked etc, all free, also walks and little beaches within the park.
I can think of others but maybe later...closer to the time you could check out the Eventfinder website www.eventfinder.co.nz/, it lists kids events, free events etc as well as concerts, theatre etc etc.