We have just got back from Sydney - there for a week then went over to Manly and stayed there for a week then back to Sydney - been a few times.
You must not miss Manley, go over on the ferry from near The Opera House, takes about 25 mins.
Bridge Climb is an expensive touristy thing to do - go up the "Pylon" on the Bridge - brilliant views from there and there is a small museum inside it. A few dollars to go up it.
We ate at the fish market behind Darlinghurst Harbour a couple of times.
Blue Mountains another "must do" either with a tour or you can do it on your own on the train and catch the circular bus when you get there.
Paddy's Market near to China Town. Walk through Queen Victoria Building - near Town Hall and very big Woolies - fantastic shops for looking in, fabulous building.
Bondi Beach - (of course) - go on the train then catch the bus at the terminus - do the Coogee coastal path walk.
Walk round Kings Cross - (red light area) nice cheapish pubs and restaurants up ther
Woolworths is a food Supermarket - the other one Coles.
Australia in general and Sydney especially is now very expensive for us Brits with the rubbish exchange rates. You can spend shed loads of money or do things on the cheap (like what we do

) there are lots of nice place to visit which don't cost a fortune.
You will love it - that is why we keep going back.