Do I need a multi-entry visa for multiple trips to Australia during my Non-O visa in Thailand?

Reentry permits are obtained at immigration offices. There is also a reentry desk at the airport just before passport control.

Brandon has answered your question perfectly but it does highlight the need to use the correct language.

That’s where a lot of confusion stems from

You won't be getting any visa, because you already did that. You entered on the retirement visa, that's what you got from the embassy.

Once you are in Thailand, you will be applying for extensions from now on, not visas. You'll never have another visa again.

Look in your passport, and you'll see the stamp they gave you when you entered. This is your permission to stay (not your visa, that's gone). It should be valid for 90 days until a date. Let's use April 19th for example.

If you leave Thailand without a re-entry permit, that stamp in your passport is now VOID. But if you purchase a re-entry permit, that re-entry permit will also be valid until April 19th, and will let you leave and return. When you return, you'll be stamped back in on your old stamp, until April 19th. A re-entry permit never adds more time to your permission to stay. Unless you are planning on leaving and returning at least 3 times within these 90 days, it would not make any sense to purchase a multiple re-entry permit. You can get a single re-entry permit.

Then after your get your 1-year extension (not visa), you can then apply for a multiple re-entry permit, which will be valid until the end of your extension. Then you can come and go as you please without losing your extension.