6 nights is a lot in Sydney if you're just staying in the city, but if as you plan, you're willing to travel a bit there's plenty to do. Blue Mountains is nice, and a straightforward drive from the city, or alternately not hard to get there by train. It's quite touristy and I haven't been there for 10+ years, but I've been there probably half a dozen times and it's a beautiful scenic area. If you're into bushwalking there's plenty of beautiful walks from easy to quite physically challenging (I'm not talking rrock climbing, but long walks down into valleys then lots of stairs on the way back up) if that's your thing.
In the city the Harbour is very nice, the rocks area at the foot of the harbour bridge on the south side is good for a morning stroll around the markets on the weekend (http://www.therocks.com/sydney-Shopping-The_Rocks_Markets.htm) can grab some breakfast lunch at one of the many places down that end of town. The Bridge Walk is quite spectacular, and even if you're a little wary of heights it's not too bad. Pretty expensive now from what I'm told, but it's something that you'll remember for a long time, so is worth doing. http://www.bridgeclimb.com/ It'd be a different experience to do it at night, but I'd recopmmend doing it during the day. It's a little cheaper during the week, so book if for a clear week day.
You can catch a show at one of the many theatres around the city. They're largely all within walking distance if you're staying in the city. If you're into eating there's quite a few good restaurants in varied places around the city. If you're into Italian food Leichhardt has an Italian forum with plenty of places to choose from. You'll need to get a bus from the city, or drive if you're hiring a car, though it's not far at all. China town in the city has some good places to eat, but can be hit and miss in terms of value and quality. There's a great Vietnamese place in Surry Hills (not far from city centre) called Red Lantern, can definitely recommend it, though it's a little pricey. There's some decent places to eat at darling Harbour as well, but they tend to be pricey. http://www.darlingharbour.com/sydney-Accom...Restaurants.htm Chinta Ria is a nice Asian Restaurant there though last time I went there was a little bit of a wait for a table and you couldn't book.
The casino (Star City - whicch is going to be rebranded the Star) is not too exciting. There's a few good restaurants there and a decent buffet. They also have a theatre there so it's a nice self contained place where you can grab dinner catch a show and have a bit of a gamble afterwards if that's your thing. Maybe watch a Toon game on the big screen If you wanted to watch some football late at night in the city there's loads of places to do so (Premier League games are generally on anywhere from 9:30PM to 6AM), Cheers Bar on George St (pretty much the main street of Sydney) is not bad and gets pretty packed when there's good matches on.
You should also visit the Royal Botanical gardens in the city, has some nice views of the harbour and is a good place to partially escape the noise and hustle of the city.
You should also catch a ferry from Circular Quay to one of several destinations - Manly beach is touristy, but not bad. I'd recommend Freshwater Beach over Manly though you can cruise by Manly on the way to Freshy as it's just to the North. You could also catch a brief ferry to Fort Denison (on Sydney Harbour), the ticket includes a guided tour of the former penal site come Fort. You'll be able to see Fort Denison from the Botanical Gardens.
Bondi beach is of course a very popular tourist destination, and you might want to go there just to say you have, though you'll probably have a better time at other beaches around the city.
If you're into wine (I drink it but am not an enthusiast) the Hunter Valley is a couple of hours drive from Sydney. I've not been there so can't really comment.
I've been to Adelaide a couple of times, but it was a long time ago when I was a kid. Not a bad place, much smaller than Sydney and Melbourne but there-in lies some of its charm. There's not a heap to see in Adelaide itself, check out the CBD... The Barossa Valley is only 45 minutes drive from Adelaide and I'd say is the most influential wine producing region in the country. Given you'll have a car already I'd probably shave a day off Sydney and give yourself a day to see the Barossa instead. There's also some nice historic coastal towns an hour or so south of Adelaide (Victor Harbour, Strathalbyn) where you could spend a relaxing day.
I've done the great ocean road drive, but only as a kid obviously not driving, so don't remember the specifics of it, just that it was quite picturesque. In South Australia (the State where Adelaide is) near the border with Victoria there's Mount Gambier which is a nice region with a picturesque lake as well as caves. The drive from Adelaide to Melbourne takes about 8.5 hours with limited stops. So if you split that in two by staying somewhere around the middle you should be able to have plenty of stops where you please along the way.
Melbourne's quite a nice city, definitely has a different feel to Sydney, but that could be because I'm from Sydney so it feels more like home (though I don't live anywhere near the city). The theatre life in Melbourne is a little better than Sydney IMO so if that's your thing there should be a fair few options. There's also loads of restaurants in Melbourne with some particularly popular places being Lygon Street which is packed with Italian restaurants and cafes. There's some good Asian restaurants in their China Town district along Little Bourke Street. The streets in Melbourne are easy to navigate and they have a tram system that will get you around pretty easily. There's a few tourist trams that run a set circuit that you can hop on and off for nix. Shopping is apparently pretty good in Melbourne, but that's definitely not my thing. The Bourke St Mall has the big department stores (David Jones and Myers) more exclusive labels can be found on Chapel St. Melbourne is a cafe city, with plenty of places to stop and grab something to eat with a coffee. A good breakfast at a cafe can set you up for the day.
I'd probably spend a night extra in Melbourne and shave one off Brisbane..
In truth I don't know much about Brisbane. Have been there but only on the way through to go elsewhere. Don't think anything in particular stands out as something to see there, so I'd chop it down to one day there on your way to the Whitsundays.
I've not spent enough time up there to comment on this part of the country.
Apart from the Great Barrier Reef and Great Ocean Road trip you're really only seeing cities, so it would be good it you could get up towards Kakadu/The Katherine/Uluru/Darwin. It will be very hot at that time of year though and you'll become extremely well acquainted with loads of bloody annoying flies, though it's definitely worth it.
