You've gotta just got see an opera at Sydney Opera House. I went to see
Tosca, and it was just superb--you can get a ticket for as little as 22 bucks (under 10 pounds). Definitely worth it!
Take a ferry to Manly or Darling Harbour (don't take a water taxi, as these are very expensive).
You must also definitely explore the area known as The Rocks. It's full of charming shops and cafes, often tucked away in exquisite courtyards. It also has a fab weekend market, maybe a bit touristy but still worth a look, where excellent jewelry to be bought.
The Rocks also contains a wicked German restaurant and beer hall that serves up huge portions of bratwurst and sauerkraut.
You can also escape the city and head up to the Blue Mountains; the YHA there is excellent and central to see the Three Sisters. Also go on the cable car, where excellent views of the landscape are to be had.
Quick Tips:
Just to let you know...
When in Sydney, get a few things straight. French fries are chips. Don't ask for a hamburger with the lot unless you like everything from fried onions to pineapple on your burger. And even on the most overcast day, have some sunscreen on, or else you will get burnt.
Best Way To Get Around:
Buses are reliable, and the best thing to do is buy an all-day travel pass; it costs 15 dollars but can be used as much as you want on all trains, buses, and ferries. You can easily make up your money, especially if you use the ferry, which is great for views of the bridge and opera house.