Australia Day isn't considered a particularly special day. The indigenous population even considers it 'Invasion Day'.
Independence didn't happen until federation on January 1 1901. Unfortunately, that's the day after New Year's Eve, so no-one notices very much. :tgif:
Of the various public holidays, probably Anzac Day (April 25) holds the most significance to everyday Australians. It's where we celebrate the deaths of thousands of Australians and New Zealanders during defeat at Gallipoli in 1915.
Understanding why Australians do this is a key to understanding the national psyche. Another clue is the unofficial national anthem: Waltzing Matilda (about a sheep stealing tramp who commits suicide).