The sheer size of the Northern Territory means it is impossible to cover it all on a short visit. You will need to be selective.
If You Have 3 Days
Don't even think about trying to do more than one of the major spots or you'll be spending all your time in airports or on the road.
If you choose one of Australia's great icons—Uluru—you could fly directly to Ayers Rock Airport and spend two days there, taking a hike around the rock followed by a look at the Uluru–Kata Tjuta Cultural Centre near its base.
The next day take a sightseeing flight by helicopter or fixed-wing aircraft, then visit Kata Tjuta (the Olgas) to explore its extraordinary domes and end the day with sunset at the rock. A flight will get you to Alice Springs, where you can tour out to either the Eastern or Western MacDonnell Ranges to explore the gorges and take a dip in a waterhole.
If you make Darwin your starting point, head east early on the Arnhem Highway to Fogg Dam to view the birdlife. Continue into Kakadu National Park and picnic at the rock-art site at Ubirr. Take a scenic flight in the afternoon, then a trip to the Bowali Visitors Centre, and you can overnight in Jabiru.
On the second day, head to Nourlangie Rock; then continue to the Yellow Water Billabong cruise at Cooinda and stay there for the night. The next day, drive to Litchfield National Park and visit Florence, Tjaynera, or Wangi Falls.
The easiest way to see Purnululu's amazing "beehive" rock formations is from the air. To do this you'll need to fly to Kununurra from Darwin or Broome, then take a sightseeing flight.
If You Have 5 Days
Five days allows you to spend more time at your preferred destination, or, indeed, combine two, though it will still be rushed.
In the Red Centre follow the three-day itinerary, finishing in Alice Springs, then fly to Darwin for a two-day trip to Kakudu National Park. Alternatively, start in Perth with the three-day itinerary then add on a return flight to Ayers Rock Airport to visit Uluru; you'll be there in time for sunset before departing the next morning.
Another option is to fly to Kununurra from Perth, then take a sightseeing flight over Purnululu's rock formations, though this is time-consuming, as the flight time to Kununurra is around three hours.
You could also spend a couple of days in Perth and Rottnest, then fly to Broome, a fascinating pearl producing town, for two days.
If You Have 7 Days
Seven days will allow you to hop, skip, and jump around Western Australia and the Northern Territory. Start in Perth, spending half a day in the city, including King's Park, then take the bus or train to the heritage city of Fremantle for lunch of fish-and-chips by the water and a stroll through the heritage precinct.
The next day, fly to Ayers Rock and watch sunset at Uluru and spend the next day driving to Alice Springs, visiting the Aboriginal Australia Arts and Culture Centre before a late afternoon camel ride along the dry Todd River.
The next morning fly to Darwin and head to Kakudu National Park for two days before returning to Darwin and flying back to Perth. Alternatively, fly west to Kununurra for a night, taking in a sightseeing flight over the Bungle Bungles in Purnululu National Park before flying on to Perth; or, pass over Kununurra and fly on to Broome, where you can laze on the fabulous Cable Beach before taking a camel ride or checking out the locally crafted jewelry of pearls and diamonds. You could start this itinerary in the Red Centre, flying in from Sydney or Melbourne before returning there from Perth.
If You Have 10 Days
If you follow the seven-day itinerary, you can afford to spend an extra day in Perth, either taking the ferry to Rottnest Island, a lunch and wine-tasting cruise on the Swan River, or even a winery tour to Margaret River. You could also add in a two-day safari trip from Kununurra to Purnululu National Park, camping out in the bush camps there.