I lived in Sydney for a while as a student some years back and have been back for holidays last year.
When are you going down to Sydney?
There are tonnes of things to do, just that it is pretty weather depending. You should climb the Sydney Harbour Bridge and also go to the unique food places on trendy streets like Crown St, Surry Hills or even in South of inner Sydney. (Check out www.citysearch.com.au
for more details of each city).
My recommendation is to buy things like surf wear at Bondi, Manly Beach. Cheaper versions can be bought at Paddy's Market as well as the factory outlet above called Market City.
If you like or have time, can go to Birkenhead Point (take bus 501 from central and walk- the bus drivers will tell you where to stop if you ask them for help- very friendly!) and buy at the factory outlets. There's another at Maroubra but it's in the neighbourhood and the stuff is not all branded or that good.
Eat authentic Thai food at Newtown, Italian at Leichhardt. there is the Rocks market on Sundays at The Rocks (just 2 mins from Circular Quay). If you like swanky stuff, you can dine at the restaurants near the Opera House(the locals called it the "Toaster"). There's also a heritage walk around the Rocks and you may find some funny pubs there... it's pretty romantic in the evening around 4-5pm but try not to walk alone too late past 9pm, esp in dark areas near the Central Train Station or dark alleys.
Walk at the Domain Park which is the Botanic Gardens of Sydney, it's behind the shopping area (near George St)and is near the Contemporary Museum of Art. Beautiful view of the city (there was an Olympic Cross Coutry Run there) and you can see Opera House AND the Woolloomooloo Wharve there. You can also eat the famous hotdogs AND PIES! at Woolloomooloo...
If you're keen on nature, drive up to the nature parks north of Sydney. It's pretty exciting.
Now it'll be turning pretty cold, so I don't suggest swimming on the beaches but its a good experience lah.
You'll get to experience the usual touristy places but make sure you don't eat too much at the food courts. it's pointless, the food's almost as expensive as going to a restaurant and tastes like shit.
Cafes are excellent, not just for their $2.50 Cappucinos (even got soy milk or low fat! Favourites!) but their bistro food is really solid! I went to Sydney Uni and there's a neighbourhood called Glebe and there are fabulous cafes and restaurants there. Bus 431/433 from The Rocks, George St City or Central Station.
Some websites you can check:
I miss my time there and really wished I could go back again and again.