I haven't been to Melbourne for a long time but it is a large city, and therefore not that much different to other cities. If you're spending a week, or so in Sydney, then I would consider that I have done the city thing and look at doing some other things, which can only be done in Australia ie
I would consider doing some of the following:-
Ayers Rock / Uluru - which is about 3hr flight from Sydney. This is about a three day trip, once you have walked round / up Ayers Rock, seen it at sun rise / sunset. And had a seperate trip to The Olgas, this is probably only worthy of about 3 days.
The following are on the East Coast, running South to North:-
Brisbane - Byron Bay is a very nice beach and much nicer than Surfers Paradise, which is nearby. I would recommend Byron Bay. Brisbane is about an hour away by plane. You can swim in the sea here, but you have to be wary of swimming in the sea further north of here, due to jellyfish. Can't remember whether this is seasonal.
Fraser Island - Huge sand island, with some really nice scenery and things to see. Hire a car and drive round
Whistsunday Islands - Sail round these islands on a 3 day boat. Go snorkling / diving. Some really nice beaches.
Cairns - Go diving on the Great Barrier Reef.
Get a copy of Lonely Planet, and see which of the above are most interesting. After seeing Sydney, I would go and see somethings which are Australia specific, which you couldn't see anywhere else in the world.