If you have three weeks, I'd think you could do Sydney, Port Douglas (fly into Cairns) and Uluru. You really only need about three days at Uluru.

And you'd be doing internal flights regardless, I think -- you wouldn't want to try to do Adelaide and Melbourne on ground transport. It's doable, but it's a long long haul. Qantas usually has specials where you can buy a multi-city domestic air pass for an extremely reasonable rate. You have to package that with a Qantas flight to Australia, but why not? Seriously worth checking out.