Does anyone know if US Post ships to Australia?

Of course. The US postal service will send almost anywhere. However, they are only responsible for getting the package to the destination country. Once there, that country's postal service takes over.

Australia has a good internal postal system, and the US post office will put your item on a plane to Australia, where that post office will do the final delivery - after it passes through customs.

Package the item and take it to your own local PO. They can tell you what customs forms you need to fill out and what various types of shipping will cost you. For a little extra money, you can get a tracking number which will help, sort of, to see the progress of the package.