A really big and nice city
Besides of taking pictures of the Opera House and the Harbour Bridge, go to Blue Mountains, Royal National Park (these 2 places are really far from the CBD, but you have to go), Darling Harbour, Sydney Tower Eye and Palm Beach (walk the trail to the top of the hill and see the sunset from above). And don't forget to do the coastal walking from Bondi to Coogee.
Insider Tip:There is a famous reserve called Morisset Park where you can see kangaroos and koalas, but instead of going there, try the Featherdale Wildlife Park. You don't have to pay anything to enter the Morisset, but it'll take you 3 hours to get there, you can't touch the koalas there and if it rains your fun will be over, once it's an opened reserve. If you decide for the Featherdale, you will have to pay about AUD 25 to enter, but it"ll only take you 1 hour to get there and you'll be able to touch the koalas.