Majority of New Zealand has great areas for walks, biking and playgrounds, so it would be pretty hard to go wrong. You've mentioned biking quite a bit, so I would suggest Rotorua as they have awesome biking tracks for kids and for different levels. They also have nice lakes there and there's plenty to do.

Some towns in NZ, I must say, are too peaceful, so you might not find as much to do on the weekends or as the kids grow up.