Can I extend my 30-day non-visa stay in Thailand to 60 days, or should I apply for a tourist visa in Australia?

Get a TR visa from the Consulate in Sydney.

You will be admitted to stay for 29 nights 30 days when you enter on visa exempt (without visa for your nationality). It is eligible for a 30 days tourism extension at the local immigration office, you can stay

59 nights 60 days total. But it is only once. If you are planning to stay more than 30 days, it would be mush less hastle with getting Single Entry Tourist Visa that will give you 60 days on entry. SETV entry is also eligible for a 30 days tourism extension, you can stay up to 90days total if you need.

The Visa Exempt Scheme gives you 30 days on entry, it can be extended once with another 30 days based on tourism at a local Immigration Office, NOT on the airport.

If you plan to stay longer than 60 days, get the Single Entry Tourist Visa before you make your way into Thailand and get 60 days on entry and this can also be extenden once with another 30 days based on tourism at a local Immigration Office.