It seems the first personal computers to be launched were the Tandy TRS-80 and the Commodore Pet. The first TRS-80 computers were delivered in November 1977 and the Commodore Pet was delivered in early 1978.
Tandy opened its first Australian store in November 1973.
Tandy in Australia were especially well known for their audio and radio products, most of which carried their Realistic own brand label. Their business expanded rapidly during the CB radio boom of the middle of the 1970s, and the personal computer boom later. Their TRS-80 series of computers was particularly well known.
Dick Smith started its first store in 1968. It was a serious competitor to Tandy. Both stores had similar products for sale. It made a clone of the TRS-80, called the System 80, released in October 1981. It also made a clone of the Apple II called the Dick Smith Cat. The reference for the Dick Smith Cat claims the System 80 was released in 1980.
Either way, Tandy with its TRS-80 computer would have been the first personal computer store in Australia.
However, such computer stores were not the only way Australians acquired personal computers for home or the office.
Starting in the early 1980s and going into the early 2000s, it wasn't unusual for home and office computers to be acquired via a local OEM store. Such stores would make a bespoke "IBM compatible" computer to the specifications the customer wanted, or close to it, depending on what was technologically possible at the time. The operation system was usually MS DOS but some OEM stores suggested DR DOS.