For 1890-1960 you can try findmypast.co.uk migration search. You can see basic details but need to pay to see ship name etc, though often once you know the year you can find the full info elsewhere.
After 1960, the National Australian Archives (NAA) recordsearch site is very useful.
Lots of sites for early immigrants but you usually need to know the ship name, or at least an approx year. It helps a lot if you know which state they arrived in, i.e. South Australia, Victoria, NSW. Each have their own searchable records sites.