Woomera has been and still is the primary launch site in Australia for rockets. A recent additional site has been the Koonibba Test Range, from where,
... rocket will carry a small payload into the thermosphere where it will be released from the rocket to fall gently back to earth under a parachute ...
During the Cold War, Woomera had the second highest quantity and rate of rocket launches in the world after NASA's facilities at Cape Canaveral.
Between 1957 and 2007, over 518 launches from Woomera were recorded.
Despite this, only two satellite have been launched from Australia.
WRESAT, quickly designed and made by the University of Adelaide, it was launched on 29 November 1967. The US had been testing rockets at Woomera and after completing their tests they realized they didn't need to use the last rocket, a modified Redstone rocket. To avoid the logistics of returning the rocket back to the US, the rocket was made available to Australia to send "something" into space. WRESAT completed 642 orbits, in a nearly polar orbit before plunging into the North Atlantic Ocean, west of Ireland. It transmitted scientific data to ground stations during 73 of its orbits. The National Archives of Australia have a video of the launch.
The second satellite launched was Britain's Prospero X-3 on 28 October 1971. It was launched using a British Black Arrow rocket.
Edit 27 Mat 2021
As with WRESAT, Prospero, X3 was launched northwards in a polar orbit.