Well the zoo is closer to the city than the zoo here in Singapore! It is only the next suburb north of the city and if you are staying in the city you could get a tram up there in less than 20 minutes or a taxi would have you there in 10mins. It is flat and easy to get around the zoo.
The aquarium/underwater world attraction down at the river is good for kids.
Catch a tram out to beach at Sth Melbourne/Middle Park/St Kilda for a run around in the sand and playgrounds without the stinking heat of here. Note Melbourne is on a bay so does not have classic Australian surf beaches unless you drive out of town - bay beaches are like here with no surf/waves.
If you don't mind going out of the city there is an inner city Childrens Farm at Collingwood if your child is not used to farm animals they might find this interesting. You can walk around and feed the animals - it is a working farm on the river between Collingwood & Kew and very laid back/relaxed despite being only a few kms from the city.
Sure the tourist bureau would be able to load you up with plenty of other stuff for kids as well. Musuems etc are oten having special exhibits for kids and so forth.
Have fun!