Hmm....interesting question. Not sure if I can do it justice, since I've studied American history more than Australian. But I'll give you my thoughts, for whatever they are worth.
If I compare US and Australian attitudes (in general terms) there seems to be more similarities than differences.
Much as the Gast painting shown above seems to see westward progress as bringing light to the dark, knowledge (the book) and technology (the telegraph wire) to untamed lands, so too did Australian attitudes centre around western culture being 'better' and representing civilization. And civilization was almost universally seen as 'good'.
Indeed, the Victoria government (back before we were federated) actually used the term 'Manifest Destiny' in a document (late 1800's) in which they proposed annexing Pacific nations (Samoa from memory, might be wrong on that) in order to bring stability and economic growth to the region.
But for a variety of reasons, I don't believe we were ever going to have wholesale frontierism in the way the States seems to have had. Arable land in Australia isn't rare (compared to the small population) and it's almost entirely centered around the edges of the continent, particularly the South East. Outside of that, there are massive tracts of desert, which (to this day) have very limited infrastructure, and are very difficult to survive.
Looking at that, you more or less can draw a line from Sydney to Melbourne in the green, fertile stretch to the east, while all of Tasmania (the small island to the south) is fertile.
Rather than expansion across a map, our history was more establishment of small colonies on a repeated basis, mostly either by sea, or where gold was found. In some cases the colonies prospered (eg. Perth originated from the Swan River Colony) and in some cases they didn't (eg. Windsor), but it wasn't really possible for an individual to up and leave an established colony to travel across the country to the 'less-developed' lands in the same way one could in the US.
To me, the arguments by Frederick Jackson Turner on American individualism, and the role the frontier played in establishing it are interesting, but I suspect a little overplayed. Still, you guys have ended up celebrating a particular brand of freedom, and individuality we don't subscribe to here, to the same degree. Perhaps our unique geography forced us to rely on mates through our inception?
