You're not seriously thinking of driving, are you?
The most direct route is 1040 miles, so that would take around 21 hours of driving at an average of 50mph. So that would be 3 days of doing nothing but driving and taking meal breaks. To break up the journey, and to include some sightseeing, you'd really need to allow a full week. However that's the inland route (which goes nowhere near to any beaches).
Travelling on the inland route to Sydney, but then following the coastal road to Brisbane extends the journey to 1130 miles so you're still realistically looking at the best part of a week for a leisurely journey with some sightseeing and beach fun.
I'd suggest flying from Melbourne to Sydney. (Jetstar, Qantas, Tiger Airways & Virgin Blue operate on that route). Spend a few days finding out what the city has to offer::
http://gb.sydney.com/
Then you can either fly to Brisbane (with the same airlines) or drive the coastal route. That's still a 575 mile drive which, with some stops on the way for sightseeing etc, would still need several days.
When considering distances in Oz, the following might help you to understand the scale of the place. Take a look at the whole of the EU
http://www.worldlicenseplates.com/maps/EU.gif
and think how long it would take you to drive between Greece and Ireland, or between Finland and Portugal. The area of Australia is 75% bigger than the whole of the EU put together!
Chris