The first major routes across Australia were associated with songlines, or dreaming tracks, passed down through the generations from Indigenous Elder to Elder.
This knowledge - conveyed through songs and other expressive traditions - described the adequately navigable route from one distant place to another, using the land or the sky as reference points along the way.
These songlines criss-crossed the country and some can be extremely long. One example was 3500km from central Australia to what is now Byron Bay.
These songs also crossed the language barrier (there were an estimated 250 Indigenous languages at the time of British invasion, and 800 dialects) with different parts of the songs being in the local dialect of the Country which they traversed.
These songs also tell stories that go far beyond mere navigation and are in fact deeply rooted in Aboriginal culture.
Meanwhile some of the songline paths are now paved as major routes today, such as Adelaide to Perth or Darwin to the Kimberleys.
With the introduction of colonisation, horses and then vehicles with wheels (carriages and stagecoaches), the first European-style roads started being built in 1788.
The very first efforts weren't exactly Roman-type feats of engineering though. They were just cleared paths with no drainage or hard surface.
There was no system-wide planning implemented either, until Governor Macquarie arrived in 1810. His term went until 1822, by which time there were three major routes, with the one heading west to the new wheat-growing settlement of Bathurst considered the most important.
The network has been very gradually expanded and improved ever since, with each new state and territory eventually doing planning of its own, until finally there was a national network of route numberings in 1955, and the federal government taking responsibility for funding the most important links, introducing the (now-superseded) National Highway system in 1974 (although some remote sections at that time were still dirt tracks).

Early on though, one such improvement to the infrastructure was Lennox Bridge in Blaxland, NSW constructed from 1832-33. Heritage listed in 1999 and still in use today, this was the first (therefore also making it the oldest surviving) stone-arch bridge on the Australian mainland.
It was built to provide an alternative climb westward that wasn't as steep or sharp or dangerous as the perpetually damaged switchbacks of Bathurst Road (now Old Bathurst Road) between what is now Emu Heights and Blaxland.
Lennox Bridge appeared to be an exception to the general underfunding the system suffered. This was exacerbated greatly as traffic increased, and that increase was dramatic during the gold rush of the mid 19th century.
Local governments (mostly councils today) were introduced to fund, construct and maintain the roads in their area. Most of these were formed in the 1860s and 1870s. As if to emphasise the point that their primary function was roads, it wasn't until the 1970s under Gough Whitlam that local governments could receive federal funding grants for things other than roads.
Additionally, federal funding for roads didn't come into place until the 1920s. Even then, the The Public Works Act passed in 1922 was conditional on the states matching this funding pound-for-pound.
With the increasing prevalence of motorised vehicles in Australia, state-based road authorities were established from 1913 to 1926. It's a system we're familiar with today, with the states taking responsibility for the main arterials, and local governments still in charge of the other (non-federal) roads.
Road rules are also the responsibility of states and territories, while the design standards of new vehicles is left to federal legislation (the Australian Design Rules).