DIRECTIONS
By plane
There is an airport in Alice Springs. Currently Qantas has connecting flights to Darwin, Sydney, Melbourne, Brisbane, Adelaide, Cairns and Perth. There is a flying shuttle to Uluru but Uluru has an airport (Yulara) too, so if you're just flying in to see the rock, you don't have to stop off at Alice. (Although you should!). Tiger Airways is by far the cheapest way to get to Alice Springs. They only fly from Melbourne three times a week at the moment, and often have fares as low as $80, although sometimes as cheap as $9.95, so if you're looking for a cheap flight there, check the Tiger site every day until you find one!
By train
The Ghan is as infamous as the Orient Express, a long train ride over a large land area, and got even longer in 2004 with an extension right through to Darwin. Don't expect complete luxury on the Ghan, however. The rolling stock is rather dated, and while adequate, it was purchased used, and has not been highly refurbished. The scenery is nice though. Expect to pay a premium over the airfare.
By car
Alice Springs is 17 hours drive from Darwin, and 18 hours drive from Adelaide. The Stuart Highway from Adelaide is well-maintained and goes right through Coober Pedy, an underground town famous also for being the opal capital of the world (and therefore well worth stopping off for a visit on the way). It continues through Tennant Creek and Katherine all the way up to Darwin. If you are driving on the main routes from other Australian Capital cities, you will likely not be driving on remote roads, and provided you do not take side trips, there will likely be passing traffic. It is worthwhile reading the safety tips for Driving in Australia and always carrying water and ensuring you know the location and opening hours of your fuel and food stops. Read the rental car conditions carefully. Rental cars in Alice Springs do not offer unlimited free kilometres. Rental cars hired outside of the Northern Territory may not be able to be driven into it. Driving after dark outside of the city limits may be prohibited.
By Bus
Greyhound ply the route to Alice Springs from the north and the south.