Wow John what a trip I agree with the below answer and all the places listed. The Great Barrier ReefUluru-Kata Tjuta National ParkWhitsundays and Fraser Island are all must sees. I would stay in the north east depending on how you plan to travel. Flick me an email if you want some more detailed advice. We just set up a bespoke travel agency and would love to help you with ideas.
3 weeks is a lot of time, you have plenty of options. If you're looking for a detailed itinerary, you might want to contact a travel agent that specializes in Australia. Otherwise, if you're exploring on your own, there are two main routes that might be interesting. The first would be to go from Darwin to Cairns and Port Douglas to see the Great Barrier Reef first, then continue down to Sunshine Coast (Queensland) and Brisbane. You could also take a Southern route straight down through Alice Springs to Adelaide, hopping over to Kangaroo Island, then continuing on to Melbourne and visiting Tasmania, finally up through Sydney and to the Gold Coast and Brisbane. Really, with short domestic flights on Qantas or another airline, you could quickly get to each major city and spend 3-4 days at each stop to explore different areas. A travel agent could suggest specific hotels and attractions and customize it to your preferences.