So it depends how long you have. I’m going to imagine you’re coming for 3 weeks. I would do the following;
1-4 Arrive Sydney. Recovery day, walk to opera house, botanic gardens, chill and enjoy being somewhere new. visit the Rocks, do a walking tour. Hyde park barracks is an excellent way of finding out the history in a child friendly and interesting way. Do an open top bus tour. Do the Bondi to Coogee walk (about an hour and a half).
Day 5 -9 fly to Cairns and pick up a car. Drive to Port Douglas. Stay in Port Douglas. Go to Cape Tribulation. Do an organised tour. Sky rail to Karunda, Hartley’s Animal Park. enjoy the rainforest and beaches. Do a Great Barrier Reef trip.
Day 10 -13 fly to Darwin. Trip to Kakadu, organised couple of days there.
Day 14 -16 fly to Uluru, see Uluru, organised tour, see it in the morning and evening. Lots of organised short tours there.
Day 17- 20 Fly to Brisbane/Gold Coast. Couple of days, go to Australia Zoo, the Glasshouse Mountains, theme parks. alternatively, fly back to Sydney and have a couple of days there - Taronga Zoo, go to a sports event, walk over the bridge, Maritime Museum.
Fly back from either Brisbane or Sydney.
I would recommend going in April\May or September/October.
if you go in December/Jan the top end (Queensland/NT is boiling and humid and you also can’t swim in the sea). July/August it’s pretty cool in Sydney but fine with a coat. Normally dry (although wet this year!)