Based on its Scottish origins, the first mechanics' institute to be established in the Australian colonies was formed in Hobart in 1827. Dr James Ross, from Aberdeen University, lectured to the ambitious young men on topics which included engineering, mechanics and steam engines. Within a few years, in 1831, classes were held on board the Stirling Castle for 52 Scottish mechanics travelling to settle in New South Wales. Scottish Presbyterian ministers and teachers emigrated with the group, loaded up with workbooks on economics and mathematics.
Sydney School of Arts, Pitt St, opened 1833.
I am indebted to Joan Beddoe’s paper Mechanics' Institutes and Schools of Arts in Australia for the information. The Sydney School of Arts was formally established in 1833, the aim of the new institution being to provide further education for working men through public lectures, classes, and the establishment of a library. Major Mitchell, the Surveyor-General and well known explorer, was elected as president of the committee, whose first task was to find a suitable site. Library facilities were to be made available, along with activities such as lectures, debating and essay writing, provided that the topics did not involve politics or religion. 250 lectures were given in the first 10 years on topics covering engineering, mechanics, natural sciences and the arts. Note that very little attention was paid to the science of agriculture, even though the economy of the colony was almost entirely agriculture-based.
Kogarah School of Arts, former
Sydney - City and Suburbs blog has a much better photo of the Kogarah School of Arts which opened in 1887. The two storey school was used for various art classes and technical education while the hall at the rear was used for meetings and local cultural productions. The good citizens of Koragah may have made the building more beautiful and Italianate, but the goals of this School of Arts were identical to any of the Mechanics' Institutes found in Victoria at the same time: to raise the moral fibre of society via educational classes and cultural activities; and to help working men to acquire vocational skills necessary to improve their working lives.
I still have no idea why some NSW organisations chose to call themselves Schools of Arts, rather than Mechanics' Institutes. The School of Arts in Young, Newcastle School of Arts and Clifton School of Arts chose one classification; The Sydney Mechanics' School of Arts and The Lawson School of Arts and Mechanics Institute chose both; many of the others simply called themselves Mechanics' Institutes.
Another useful reference is Pioneering Culture: Mechanics' Institutes and Schools of Arts in Australia by Philip C. Candy. Note that Candy and Beddoe used used both terms in their book titles.