The first known people to have settled in Australia are known as the aborigines. They are known to have been on the continent since the late Pleistocene epoch (~30,000 years ago). This migration is around the same time as movement of Asians to North America via the land bridge. Australia was likely colonized via sea navigation or possibly as a result of a sea navigation accident. The exact origin of the aborigines is unclear. However, during the initial migration, sea-level was much lower (~200 m) and the Australian continent was 25% larger. In fact, during the late Pleistocene, it would have been possible to walk from the Australian continent to Tasmania or New Guinea on dry land. Dingos, or Australian wild dogs, are one of the lasting biological remnants of the aboriginal migration.
The first Europeans to come to Australia came with Captain James Cook in April of 1770. Cook left England in August 1768 and returned in 1771. This was not initially a mission to find Australia; it was a mission to Tahiti, which is in the middle of the Pacific Ocean, to observe the transit of Venus across the Sun. Then after that mission was accomplished, Cook opened sealed orders that instructed him to look for a hypothesized continent that scientists were calling Terra Australis. So Cook convinced a Polynesian named Tupaia to come on the ship with him, and Tupaia helped Cook to navigate the southern Pacific. They completely circumnavigated New Zealand and mapped it in complete detail in 1770. The Cook party sailed across to Australia and encountered aborigines in Botany Bay (named for the unique botanical specimans). Botany Bay is just south of modern day Sydney.
A quick note about Cook, he had two more expeditions, among which he nearly went to Antarctica (sailing into the Antarctic circle), he “discovered” Hawaii, and he was the first person to map North America from California up to Alaska. He was killed by a group of Hawaiians.
In 1787, Australia was became an official penal colony of England under King George III. And in 1788, eleven ships landed in what is currently known as Sydney harbor. In all 160,000 convicts were sent from England to Australia.