You must have a valid U.S. passport and a visa or an approved Electronic Travel Authority (ETA) to enter Australia. Most U.S. passport holders traveling to Australia for tourism or business purposes for less than 90 days can obtain an ETA.
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Unless you are an Australian citizen, you will need a valid Australian visa to enter the country. New Zealand passport holders can apply for a visa upon arrival in the country.
A US green card not only lets you reside in the US but also provides you with VISA-FREE access to many countries and territories. Currently, there are about 38 VISA-FREE countries for US green card holders. A US green card is a pathway to a US passport.
There is no 'minimum' bank balance that you need to show for an Australian tourist visa. What you need to demonstrate is that you have enough funds to support yourself during your trip. A one week trip staying with your brother is going to cost a lot less that a 6 month trip travelling around all of Australia.