My girlfriend and I want to go on vacation to Europe, but she does not have a US passport or a passport of any of the EU countries. She holds a Belarus passport so she would normally need a visa. Does having a green card eliminate the need for a visa, as it would if you had a US passport?

I’m retired, and looking to stay awhile. I don’t plan to work, just enjoy Aussie stuff. Anything I need to consider? Do I need a real visa, or can I just stay for three months on my US passport?