I'm in Australia and I'm looking at US hosting options because the prices here are embarrassingly extortionate. Are there any good, free online tools I can use to find out which location in the US has the fastest performance - in terms of latency and download speed - for connections from Australia?
SnapOverflow
The US is about as far from Australia as you can get, and so no matter where you go you're gonna have latency issues. I don't know if the rates will be any better, but try looking in Japan, the Philippines, India, or just about anywhere else (aside from Canada).
A bit portion of this will depend on carrier, but in general, you will want to look at hosting on the West Coast of the US. Try Los Angeles and San Francisco Bay Area. Given that you're looking at a big latency hit anyway, there are some lower-cost options in cities like Dallas where the colo space may be less expensive at a marginal effect on latency.
You can start here for general info: http://www.verizonbusiness.com/au/about/network/latency/
I work in the financial sphere, so most of the route/latency data I have is specific to super low-latency (read: expensive) data lines for trading systems.
For testing purposes, mtr is a pretty fantastic tool for determining latency and such.