Remember, if you travel into an Aboriginal Community, you are not allowed to take alcohol or pornography. There are severe fines if caught. Also, when visiting Arts Centres, do not travel into residential areas these are well sign posted.
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 (http://www.tigerairways.com.au) is by far the cheapest way to get to Alice Springs.
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 bus
Greyhound (http://www.greyhound.com.au)] ply the route to Alice Springs from the north and the south.
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 worth stopping off for a visit on the way. It continues through Tennant Creek and Katherine all the way up to Darwin.
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. Cars rented locally in Alice Springs usually 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.