In terms of physical proximity to the night sky, Sydney Observatory is a pretty good starting point. One of the highest points overlooking Sydney Harbour, its building houses three telescopes — including the oldest working telescope in Australia, which was built for the 1874 transit of Venus. And for those of you who prefer gazing at the star closest to us, without getting completely blinded, there's a telescope that lets you look at the sun.
The Observatory runs night and day tours, workshops exploring early Indigenous astronomy and a range of short courses. Plus on those special days when there are eclipses, the Observatory hosts dedicated events. This is definitely the first step for every would-be Galileo.