I am in no way affiliated with this outfit, but I did go walkabout with them in 2003 for a day. No joke - it was a life changing experience. The guy who runs it is named Evan Maru. When I met him, he was still in graduate school; he's half aborigine himself, and was writing his thesis about his own tribe, which has all but vanished (being so near to Sydney). Really glad to see him still offering tours a decade later.

http://www.tourstogo.com.au/tour/611-blue-mountains-walkabout/

Blue Mts are spectacular (re: prior reply) but if you'd like to see iconic sights of Sydney, here's what I'd do: - Opera House. Just walk around, check it out, enjoy the waterfront in the sun... grab a meal or a drink at the Opera Bar with stunning views of both the Op House and the Harbour Bridge - Continue a waterfront walk from the Opera House through the botanic gardens and Mrs Macquarie's chair. Lots of history and uniquely Australian / Sydney - take the ferry from downtown to Manly: it's a cheap tour basically but gives you spectacular views of the city from the water! Manly is a cool little suburb/neighborhood with an ocean side (surfers!) and a harbour side (boats!)- lots of beach to kick around in. - If you're athletic, an alternate to the ferry is to do the Spit Bridge to Manly hike. Trails along the bushland (that's woods to us Americans) and ridgetops of the north side of Sydney. Amazing views of the city, historic aboriginal carvings, lots of Aussie flora & fauna. - Skip Bondi Beach. It's really overrated. - The rest depends on what you're into. The modern art museum or Australian (natural history) are worthwhile IMHO...coffee here is amazing, find a cool cafe in hipster newtown or surry hills...walk Bondi (I know I know it's still overrated) to Coogee beach for beautiful coastal views and some nice interpretave signage...