Sorry for the wiki links, but this is massive question...

There's quite a lot of gang activity in the 1920s, the most famous being the Razor Gangs of Sydney, which draw their name from the weapons they used. They ran cocaine, and operated speakeasys. Most of Australia had 6 o'clock closing, where the pubs would shut at 6, and so speakeasys became common after this. A couple of the razor gangs were also run by women.

In Melbourne the most famous gangster of the time was Squizzy Taylor, who was involved in similar kinds of things - there's a fictionalised TV show about him, called Underbelly, Squizzy... Other seasons of Underbelly outline the recent gang wars in Melbourne...

Before that, and the only one really tied directly to convicts, would be Bushrangers who with a variety of motives and styles, held up people particularly around the goldfields from the 1850s to the 1890s... A few of these were previous or escaped convicts, but others were not... The most notable would be Ned Kelly (who was native born, and wore a suit of armour in his final shootout), as well as Captain Moonlite, Captain Thunderbolt, Gentleman Ben Hall, and Captain Melville...

In terms of convicts in general, there were around 162,000 sent out between 1788 and 1868, whereas in 1852 alone, 370,000 free settlers arrived, and the population was over 3 million by the 1880s... So there's no real continuous 'criminal legacy' The US had around 40-60,000 convicts sent out, and I doubt you would see much of of a legacy there...

http://en.wikipedia.org/wiki/Convicts_in_Australia

http://australia.gov.au/about-australia/australian-story/convicts-and-the-british-colonies