Beauty.... Epic it is....
Firstly, can you hold off until May?
Epic, with great scenery and remoteness = Fly into Alice Springs, head out to Glen Helen and Mt Sonder, there are a few Gorges / waterholes to visit on the way. There are plenty of cars travelling this route, well, plenty for OZ, maybe not what you're used to! This section is tar so it's easy pedaling and will get you into the groove of being out on your own. I have a mate in Alice who'll sort things and give you the lay of the land.
From Glen Helen you keep heading West, through Tylers Pass to Gosses Bluff then South to pick up Larripinta Drive where you turn West to get around the Gardiner Range and into Kings Canyon. (bit of a resort and campgroud there. That section is all dirt. From Kings Canyon head down to the Lasseter Highway, turn West and head off to Uluru (Aryes Rock) and Kata Tjuta (The Olgas), maybe doing a little detour to climb Mount Conner.
That ride will be approx. 650 to 700Km's. You'll need to carry at least 15 liters of water, this will give you a comfortable 3 days, and 4 days of water if you have to ration it. Free camping is the norm, paying at resorts is an option (they're good for showers!) Temps in March range from (absolute 14 to 45 C), (long term ave: 17 to 32 C) and in May (absolute 8 to 38 C) (long term ave: 8 to 23 C)
Having scooted around and climbed the Rock and the Olgas, and if the budget can stand it, a flight over them, puts the whole place into perspective! jump on a plane and head off to Perth for the beach and maybe Rottnest Island and the Fremantle coffee strip. There's also a few trails in the hills that you could ride if you wanted. For this route, flying in via Asia through Darwin and back out via Asia may be cheaper. An alternative is to fly out of the Rock into Brisbane and then into Byron Bay for the beach, it's more beach friendly there in late May early June. There is a farm stay that has a MTB circuit, google Byron Bay Farmstay. For this route, in and out of Sydney may work out cheaper. Of course, if you had the time, there is always the option of riding back to Alice via a different route and flying out of there.
OK, that's an epic, and one I doubt you'll ever regret.
Let me know what you think. Don't get spooked by it, I've ridden lots of remote stuff and can give you heaps of advice, and as I said, my mate in Alice will bring you up to speed, but if it's too much there are other rides.
Al.