I'm guessing, based only on the fact that this question was posted in the Sydney forum, that you're only going to Sydney?
There are some lovely beaches around Sydney, which will be quiet, but by no means deserted, in June. I suggest you catch the ferry to Manly and kill two birds with the one stone. The views from the ferry are fantastic.
Tooronga Zoo is pretty good, from all accounts.
Check out the Blue Mountains to the west of Sydney for some beautiful scenery.
I don't need to tell you about the Opera House or the Harbour Bridge, do I? They're iconic landmarks of Australia.
If you're planning to go elsewhere in Australia, see the Barrier Reef in Queensland, where the weather should be quite mild in June. Or you could rug up and see the Penguin Parade at Phillip Island a couple of hours out of Melbourne. Perth has some of the best beaches in Australia and a laid back atmosphere that comes from being one of the most isolated cities in the world. Tasmania has some breathtaking scenery.
Just remember that Australia is HUGE. It's about the same size as the United States. If you want more specific suggestions, post another question letting us know where you're planning to go.