I got back about six weeks ago from this three-day trip to the Red Centre:
http://www.adventuretours.com.au/aus...original-95519
It included Uluru (full base walk), Kings Canyon (rim walk) and Kata Tjuta. I had seen a bit of that in May 2014 (thanks, FT OzFest!) but wanted to do more. It would fit your hiking/outdoor objective, and it's hard to see much of that area without someone to drive you around as the places are far apart. I think something like this would be worth five or so days of your time.
Adventure Tours isn't the only outfit that offers similar tours. They're about midrange.
Wayoutback is more toward bare-bones;
AAT Kings is more luxurious, and there are others. Shop around to find one that hits your sweet spot. (I have no financial or other interest in any of them.) You can add a day to see Alice Springs, but I had already done that and was on a tight schedule.
Other than that, it's hard to go wrong. If you go to the Barrier Reef, don't miss the rain forest; and while you're in Sydney, climb the bridge.