Sydney is my centre. I was born and raised here and am quite attached to this city with its many contradictions. Dman summed it up pretty well in Sydney: my favorite city but I'd like to amend and add a local's perspective.
Most people will think of the iconic Sydney Opera House or Sydney Harbour Bridge (aka Coathanger) or just Sydney Harbour when they think of Sydney.
The Harbour itself has been described as one of the most beautiful natural harbours in the world, although it is by no means the largest of the lot. Although the Harbour was the site of Sydney's original colony, it was not the original landing site of the First Fleet in 1788. Instead it was Botany Bay, a large calm bay fed by Georges and Cooks Rivers, approximately 7km South of the Harbour and the CBD. Botany Bay was first chosen as the landing site by Captain Arthur Phillip because it was near Captain James Cook's original landing site at Kurnell (1770), but it was rejected becuase there was no fresh water nearby. Phillip decided to move further up the coast and found Sydney Harbour, aka Port Jackson and Sydney Cove, freshly watered by the Tank Stream (now all but disappeared under concrete and steel).
The First Fleet and its cargo of convicts (mostly petty criminals) settled at Sydney Cove and became the penal settlement known as Sydney, with Cpt. Phillip as its governor. In doing so they displaced the Koori and Eora, local tribes.
Until very recently a replica of this early colony could be found at Old Sydney Town, 250km North of Sydney on the Pacific Highway (Route #1), but it was closed for lack of interest in 2003. Sorry about that.