I too am from the outer east of Melbourne and have young children, with the oldest being 6.
I would recommend trying, Wilsons Promontory first.
It's a lovely place which attracts many people - so your child may find another to play with and to meet other happy campers.
It has all the mod cons (toilets, showers, open air cinema, general store), so it will suit less experienced campers and non-campers.
There is a lot of wildlife to wow the family including Kangaroos, Emus, Wombats, Birds and Deer.
When I was at The Prom last we went Wombat hunting with 2 other families. We found one in a couple of mins and the kids asked if they could pat it. I suggested that patting the Wombat was not a good idea - so all the kids immediately jumped forward and starting patting it. The Wombat didn't seem to mind - but it was covered in fleas.
It's also a short trip to the beach.
Other places nearby are The Upper Yarra Reservoir near Warburton and King Lake. If you don't mind travelling then try The Grampians and Mt. Buffalo.