Melbourne is a great city to visit. I've had some good meals in the whole area by Crown Casino. As mentioned earlier, there is the MCG (Melbourne Cricket Grounds) - not sure if the time of the year is right to take in a cricket game, and I think footie is done also, but still there may be things going on. There may be some athletics going on. I made it there for the Commonwealth Games, and it was a great venue.
To see the entire city, you can go to the top of the Rialto Tower. You can also ride a tram around the central business district for free.
Here's a bit more abstract of a suggestion... Make sure you get to a local grocery store. You'll want to get some Arnott's Shapes, some Big M flavored milk (the banana milk is awesome), some Violet Crumbles, and some Cadbury chocolate bars (their Cadbury is different than ours). Also, you'll want to get a good meat pie somewhere. Not one at the cricket grounds or a venue, but rather one from a bakery type of shop.
You can go see penguins at Phillip Island also (good to do at night), about 1.5 hours away I think. If you do the Great Ocean road, get you may want to take a few days to do so, and stay in some of the resort towns along the way. Then you can get out to the Twelve Apostles area (really cool rock formations in the ocean) which is about four hours from the suburbs.
I'm not biased towards Melbourne or against any of the other places, but it's the only place I've been there, so it's what I know. It's a city with suburbs. Seeing Alice Springs, Queensland, etc. are completely different though, from what I understand.
If I can think of anything else, I'll let you know.