How did Australia go from a prison colony to a first world country?

There are three things really at play in the very quick evolution of Austalia from penal colony to colonial center. First off, the penal colony existed to serve a purpose, it was not just a place to send debtors from Great Britain, it was a source if labor. The convicts in the penal colony could be deployed to work on government projects like roads, dykes, or other irrigation or landscape projects. They could also be leased out to work in one of the other industries that took off in the colony.

Secondly, Australia is rich in minerals, so it should be no surprise that mining was one of the other forces propelling a quick shift in Australia. An extended Gold Rush beginning in the 1850s brought miners, migrants, and capital to the antipodes. Related to this, access to capital and cheap labor also propelled the expansion of ranching and plantations in Australia. The late 19th century saw substantial sugar and cotton industries pick up, and right around the time the penal colony was brought to a close (1868), the labor supply shifted towards the recruitment/black birding of tens of thousands of islanders from across Melanesia to work in the fields.

Finally, Australia enjoyed a key strategic position as the British empire expanded across more of the Pacific making notable claims to Aotearoa (NZ) and Fiji, both of which shipped a great deal of their colonial produce through Australia at various times. And even further expansion into the Solomon Islands and then Papua New Guinea and former German territories after WWI. Australia was also a key shipping stations for the mining industry on New Caledonia and the plantation and mining interests in the New Hebrides/Vanuatu.

All of this is to say, it made the transition very quickly because of a large free settler population, vast natural resources, access to cheap labor, and a strategic location.

*edited to fix poor mobile layout.