Further I did not quite understand why Australia just got independent.

Here's something to fuck with your teacher. We never truly got "independent". There's a whole lot of technicalities at play. Technically the Queen of England is our head of state, but technically the Governor General (currently Quentin Bryce) is our head of state and then technically our Prime Minister (Julia Gillard) is our head of state.

As for the actual history part, it really is a straight forward procedure. The Dutch arrived in North Western Australia, realised that it was all desert and there is no way to grow marijuana they decided to leave the Indigenous alone. Then the French visited after spending a bit of time in Tahiti and figured "well Tasmania isn't that great. We'll leave the Indigenous alone". Lastly, the British arrived in 1770 and figured "we could make something of this place". So they sent a lot of Irish, Welsh and Scottish prisoners here, along with some free settlers and called it Sydney in 1778.

For the next 30 years the city of Sydney developed around the area that is currently the Sydney CBD. They chose this location because it is one of the most sheltered harbours in the world. A few towns around Sydney propped up, such as Parramatta. From 1800 to 1900 Australia has a very similar history to the United States. We had frontier wars with the indigenous population, with Windradryne being one of the more well documented tribal leaders west of Sydney, and we also had an influx of Chinese migrants due to a gold rush. During this time, Australia wasn't a nation per sé, but rather a series of separate colonies belonging to Britain.

In 1901, these colonies formed a federation. This is why we aren't an independent nation. The colonies of Western Australia, South Australia, Queensland, New South Wales, Victoria and Tasmania decided it would be much easier to manage trade between each of us if we were one nation. There are other states that formed after this federation, such as the Northern Territory (which was originally part of South Australia) and the Australian Capital Territory (ACT).

After Federation, there was a massive rivalry between Sydney and Melbourne, the two biggest Australian cities, over who should be the Capital. Sydney had a larger population and was the first city to be established in Australia, but Melbourne had always been the hub of political affairs in the nation. The two cities decided that the best spot for the capital was the halfway point between the two. However, this spot had already been occupied by another town, so they moved slightly to the east and made Canberra. The city of Canberra is still the capital today and was entirely planned, to the point where Australians joke about its roundabouts and street pattern being a message to Xenu.

After this, World War I began. The British decided that it would be best if Australian troops were sent away from the main action, to a place called Gallipoli, which is where the European and Asian sides of Turkey stretch out into the Mediterranean. The British figured it would be an easy fight. It turned out to be an absolute bloodbath. It ended in a stalemate for both Turkish and Australian troops, who suffered huge losses on both sides. The day of the Australian troops landing at Gallipoli, April 25th 1915, is known as ANZAC day. ANZAC stands for Australian New Zealand Army Corps, and is widely celebrated as a day of remembrance for Australian soldiers who died in any form of active duty.

After World War One, Australia entered the Depression. Whilst other countries were cutting expenditure during the depression, Australia thought "fuck it, we'll build a bridge or two". It was an economic policy that many thought was daft but it served the country a lot better in the long run, and many parallels can be drawn between the depression and the current GFC in terms of how we handled it. Australia entered World War Two and not only did we have to send troops to help Britain, we also had to prevent the Japanese from invading Australia. The Kokoda Trail was one of the major battlefields in Australian war time history.

After the second World War, Australia seemingly opened its doors to immigration. After decades of the "white Australia" policy, many Southern Europeans, Balkans and Jews emigrated here. After a number of years these people became accepted into Australian society whole-heartedly. Many people who are my age (I'm 21 btw) don't see Greek Australians as Greek Australians, but see them as Australians. The traits that were once considered racist stereotypes became part of Australian culture, just as much as surfing culture or "bogan" culture.

As part of the ANZUS treaty, Australia was pretty much forced to provide assistance to the U.S. in Vietnam. It was during this time that a lot of social change occurred. People began to dislike the idea of Conscription and the legal drinking age in Australia was dropped from 21 to 18 because people realised "I can be forced to go overseas and die for my country but I can't buy a beer". During the 1970's the Family Law Act was also introduced, which meant that a divorce didn't need a reason such as violence, bigamy or adultery. It introduced a concept of no-fault divorce, which lead to a change in the family structure for many Australian families. Most kids born after 1990 would have grown up with or had their parents divorced.

After the Vietnam War, there was a large number of Vietnamese refugees, which is ironic given the fact of the matter. This lead to a change in immigration policy from Euro-centric to Multiculturalism. It's not uncommon now to see African refugees in cities like Melbourne, Newcastle or Western Sydney.

In 1986 we had the Australia Act introduced, which meant the highest court an Australian case could go to was our high court. Up until then you could take a court case all the way to Britain if you wanted. That is probably the biggest defining factor in our supposed "independence".

tl;dr I hope this helps, because that's all I remember from two years of Australian History in High School.