2 Nights SIN gives you the opportunity to visit this city on a morning's excursion and leave next day, or the same night.
If you leave at night you be in Sydney next morning and if you are able to sleep in the plane probably your jetlag has gone (or not, it,s very personal). Do you have a car during your Sydney days?
Leaving next morning you'll arrive in Sydney early in the evening and you better stay at a hotel for the night. You're jetlag will be gone almost for sure now.
4 night Sydney is good for a start. Take the ferry from Manly and you'll have splendid views on Sydney arriving Circular Quay.
Strolling in Sydney for 2 days and maybe take a orginized daytrip to the Blue Mountains (or drive yourselves). You can easyly spend some more days in S but remenber, it's june so it's their winter.
Flying from Sydney to BNE and stay two nights in Brisbane.
Make sure you get a ride with the CityCat on Brisbane River and visit South Bank.
Queenstreet has many shoppingopportunities.Its on the north shore. That'll do for one day.
Brisbane's hinterland and the Sunshine coast north of Brisbane is very nice too so you can spend a few more days in this area if you like.
Between Brisbane and Cairns there are many highlights too like Fraser Island and Whitsundays but it's 1700 km by road from B to C.
Palm Cove north of Cairns is a pretty place with lots of accomodation and it's a good time to visit. Visit www.ozstays.com.au and you'll see dozens of hotels, houses etc.
From Palm Cove you can make daytrips to Port Douglas (sundaymorningmarket is a must visit) to Daintree NP, Cape Trib., Atherton Tablelands, Kuranda and many more places and of course, the GBR as a highlight. You can visit GBR from Cairns and Port Douglas. PD is closer to GBR.
QLD hol. are juli 1-16 so try to be gone before that time.
Best to hire a car at Cairns Airport.