Manly and surrounds is gorgeous and close to the city, you can catch a bus there, or a bus then train. I love the Manly area, there are plenty of places to stay too. Some nice places to eat too.

If you want to be in the actual city, Darling harbour is lovely too, close to ferrys and trains and buses, which will take you everywhere you would like to go around the city and outer suburbs too. Lots of restaurants, shops and stuff like that close by (within walking distance) too.

Good luck and have fun! I love Sydney!!