I was in Sydney last year in September and had a few free days in the city. I'll give you a short breakdown of some of the things we did and my impressions. Shoot me a PM if you have any other questions. I grouped stuff in the downtown core by location:
"Left" (Darling Harbor)
Maritime Museum - Good museum. Separate $$$ for the static displays inside and the ships outside
Aquarium - Buddy did it and thought it was good.
Harbourside Shopping Centre - Fairly large mall. Has a Hard Rock Cafe.
"Center"
Inner Harbor - Nice walk, lots to see and the obvious tourist shots of the Opera House. Start at the Opera House and walk until your under the bridge or vice-versa.
Opera House - You can walk around the outside and go into the lobby for free. They also offer a tour for $$$
George Street - Lots of shops - some high end and some fairly cheap. There's a market that sets up a few days a week on George, just past Hickson Road - good tourist stuff.
Bridge Walk - Saw people doing it but didn't have the time. Pretty $$$, but very popular attraction.
"Right"
Hyde Park - Nice resting area. Popular hang out spot.
Anzac War Memorial - Within Hyde Park. Very well done.
Australia Museum - Kind of boring. Pretty standard museum stuff once you've seen the Australia portion.
Hyde Park Barracks - World Heritage Site. Basically a look back at "Australia the Penal Colony."
Royal Botanical Gardens - Nice walk. If you have time, you can get tickets for a tour of Government House, but walking the grounds is free.
Some other stuff:
Food is hella expensive. I'm talking 8 bucks for a Big Mac expensive. You want the combo? Twelve bucks. Want a Starbucks Sydney mug? Twenty-one bucks.
Wifi is terrible. McDonalds and Starbucks have free wifi, so you at least have that. Enjoy that Big Mac.
You MUST go to the Lord Nelson Brasserie. Seriously good food and good beer. Buy their t-shirt.
Australia is the only country where you can eat both national animals at a restaurant. Didn't try emu, but kangaroo is actually pretty good when cooked properly.
If you can find it, buy a bar of Darrell Lea "Rocklea Road."
We also did a day trip out to the Blue Mountains with Blue Diamond Tours. I really enjoyed it, and it ended up being a full day of Australia. Take this tour if you want to pet a Koala.
And of course, if you have enough time to get up to Canberra, the Australia War Museum is a must-see.