It's funny, our 'independence' is kinda like the concept of marriage to me. The wedding is a way of making a union formal...even though there are plenty of folks in committed relationships who stay together through thick and thin, for richer for poorer etc. without the formality of the wedding. I'm one of those people. My partner and I are going on seven years with two kids and we've had good times and bad. Getting married somehow just isn't 'us'.
Australia as a nation is for all intents and purposes 'independent' because we make our own policy both domestic and foreign. 'Mother England' no longer tells us what to do even though our official Head of State is NOT the Prime Minister...it's the Governor General...the Queen's representative. He has the power to dissolve government and in 1976, a drunken Sir John Kerr (

) sacked the government of the day and the opposition guy got the job. This event is known as 'The Dismissal' and was an very interesting chapter in Aussie politics. There are those who believe it amounted to a bloodless coup orchestrated by the CIA. I remain open minded about that.
Several years ago, there was a push for an Australian Republic. A move I strongly support. We had a referendum on it and in my opinion, it ended up as a 'No' because the question people were asked to vote on was cleverly spun to play on people's fear of unelected officials. They never once stopped to think that the Governor General (the guy with the actual power) is not elected by the constituency. This is where it gets tricky.
The model we adopt is all important. Personally, and no disrespect intended, I don't favour an American style presidency. What I see is only people wealthy enough to fund huge campaigns being elected. Nothing wrong with wealthy folks but they aren't neccessarily the best leaders. Our previous PM is from priviledge and there's an anecdote going around that he once kicked a homeless man and told him to 'get a job you lazy bastard'. A fairly simplistic view of somebody less fortunate than himself.
And thus ends the lesson on Aussie politics.
Our own dates...well most of them centre around the worship of anyone who was at Gallipoli and Australia Day in January. That's when the states united into a 'Nation' in 1901.