After you check out The Opera-House
(to the right of circular quay)

You will be close to the Harbor Grab a Fiery ride to Manly Beach afterwards, very pleasant trip with chance of seeing Dolphins on Sunny-days.

Or you get a Bus to Bondi-Beach across the road from the Harbor (circular quay)

At bondi Beach if you walk around to the right of it, the beaches get nicer and nicer further away from Bondi(tourist trap place)

There are cheap accommodations on George Street of TownHall/Central(its noisy) close to cheap eats chinatown & markets in reading centre

From Central You can Get a Train to the Blue Mountains or Southern Suburbs, Western , North Sydney or Eastern
(bondi junction, can get bus to Vaucluse/Watsons Bay Area check out how the 1% live or to Bondi for $11 small chips)

From Town Hall walking past Hyde park There's the Museum.........

in the direction of Kings-Cross If you want to see Hookers and get tattoo's of your experience in Sydney If you like that kinda thing ;)