You can totally do this in six months, I know people who have done RTW in less time than that. And sure they didn't spend ages in each place but they got some amazing experiences and memories to last a lifetime. So don't be put off by people saying it's not enough time :)
So I would start by flying to Australia (super-expensive so restrict yourself to maybe 3/4 weeks). Since you're starting in May, that's a good time for Great Barrier Reef, so why not East Coast Aus and down to Sydney? (I heard it's chillier down Melbourne and Tasmania way at that time of year.)
So after Aus, spend a good couple of months in Asia. Your money goes far in Thailand, Cambodia, Laos and Vietnam. Standard backpacker trail destinations, but for good reasons! Ha Long Bay, Angkor Wat, Bangkok, awesome.
Now it's around August? How about partying like crazy for a couple of weeks in St Petersburg, Russia? The White Nights in August are meant to be insane, and it's a good stopover to Europe, where you could spend the last couples of months. September is really mild and pleasant weather in much of Italy and Spain so you can get some beach time and culture before flying back to the USA.
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