I suggest melbourne as a start. auckland and christchurch is a longer flight so if this is the first time they are flying, then maybe get them acclimatized to flying first.
gold coast, i will bring them when they are older and can appreciate theme parks better i.e. take the rides and recognise the characters at the WB studios.
I prefer melbourne over sydney because melbourne is a much more walkable city and public transport is very convenient. day trips can be made to great ocean road, mornington peninsula and philip island (penguins and koalas!). being much more walkable means less inconvenience with prams. hope this helps.