Hi Mike,
Welcome to IIS.
The short answer is: From about this time of year (September) until around March. The further south you are observing from, the higher in the sky they will be but nearly all of Australia away from city lights save the far northern parts will provide an excellent view on moonless nights. As they are always in the southern part of our sky, avoid locations where there is substantial light pollution from a source directly south of your observing location.
You can see more in the LMC with a humble 10x50 binocular, than a 50cm telescope will show in the Andromeda Galaxy (NGC 224 or M31) from northern climes. If you have access to a large telescope here, you could spend 10-20 hours behind the eyepiece just to cover the highlights from both clouds. Triple that if you want to go "serious". You can spend a few hours alone on just the Tarantula Nebula (NGC 2070) and its surroundings. Have you considered the Three Rivers Foundation and their regular guided trips to Australia?
https://www.ozsky.org/
Best,
L.