Handwriting has changed over the years, hasn't it?

It would appear that he showed up for service with the 2nd Australian Imperial Force , lasted only 8 weeks, and then showed up for a reserve unit, 3rd Infantry Brigade.

The AIFs were Australia's expeditionary force. 1st AIF was WW1, 2nd AIF was WW2. With a term of service of only two months, it seems he didn't make it past basic training in the first go-around, for whatever reason, and thus seems not to have gone with 2nd AIF to North Africa.

3rd Bde was a reserve unit. My guess, he got the basic training, was mustered out, and in the concern period of mid 1941 reported in for reserve duty.

Obviously see Georgy's reply below for specifics on how to get details from the Aussie government.

Hello there! As your question is related to looking for identification/information regarding military personnel, our Guide on Military Identification may be of use to you. It provides a number of different resources, including how to request service records from a number of national agencies around the world, as well as graphical aids to assist in deciphering rank, unit, and other forms of badges or insignia. While the users here may still be able to lend you more assistance, hopefully this will provide a good place to start!