Darwin is a city in Australia.
From Darwin, you can really only go South. The first town you will hit is Katherine. It is advised to take the bus to Palmerston ($3.20 for a 3 hour ticket as of September 2016) and then simply walk along Temply Terrace to the Stuart Highway, about 1km away. Traffic is slow at this point and there is a set of traffic lights. You may get very lucky as there is plenty of long distance traffic going past here. Another promising spot is in front of Woolworth and the nearby petrol station at Coolalinga. There is a bus service running from Palmerston to Coolalinga. A lot of long distance travelers stop here for some last shopping before going south and usually there is plenty of people to ask for a lift down south even though it seems to be on the wrong side of the road. If you want to travel some distance be carefull about accepting short lifts and avoid Noonamah service station because you'll find mainly local traffic there and Coolalinga is by far the better bet. Try to avoid getting dropped off in Katherine if you are not going there because getting a lift from there might involve a fair bit of walking and long waits. Let yourself drop off on one of the road houses on the way instead.
Darwin is in the Northern Territory, which means sleeping by the beach is not a great idea due to crocodile danger.