How early should I apply for a 60-day tourist visa for Thailand as an Australian passport holder, and can I buy my airline ticket before obtaining the visa?

You will need to supply your flight details on your visa application

Go look up the info on the thai consulate in canberra website. Most appropriate source of info for you. I know its there because ive referenced it myself many times

Apply via Sydney consulate for Tr60 pre-approval online application. No flight or accommodation proof is required just write the flight numbers and dates and accommodation your considering.

As Brandon said its only good for entry within 90 days of issue so apply within the 90 days. Sydney are reasonably easy and only takes a few weeks lately.

Bearing in mind after approval email is received you need to send passport and paperwork to Sydney consulate so allow time for that to be returned.

If you're going through Sydney consulate apply six weeks before your planned departure date. You will be required to give your flight itinerary, so yes you can get your ticket anytime. The sooner you get it the cheaper it will be

The visa is valid for using to enter Thailand for 90 days after it's issued. So apply when your trip is less than 90 days away.

Most embassies require proof of your plane ticket as part of your application. Check your local embassy for their requirements.