Two days is barely enough to cover the city centre really. Things worth doing though:
1. Take a ferry to Manly. You get the best views of the harbour and the opera house for the price of a ferry ticket. Manly has a great beach culture and great shopping/eats.
2. Wonder through the Rocks (southern end of the Harbour Bridge) and get souvenirs and nick nacks to take home with you or just soak up the historic atmosphere. If you are visiting on a weekend be sure to visit the rocks Sunday Markets on the northern end of George st. You can also get a walking tour of the Rocks, circular quay and other parts of the city centre. Keep an eye out for a little bakery/pastry shop on the main street of the Rocks facing Circular Quay. Its the best place I know to pickup an Aussie meat Pie and other tasty lunches and sweet treats. Remember also to check out the buskers near the Circular Quay ferry wharves. Apart from the living statues you will be sure to find at least one traditional Aboriginal (in tribal attire) properly playing a didgeridoo.
3. For downtown city shopping be sure to visit the Queen Victoria Building (QVB), the Strand Arcade and the Pitt St Mall. If you don't suffer vertigo you might want to visit Sydney Tower at Centrepoint while you are there.
4. Don't forget that you can also walk across the bridge itself for free. There is a small museum in the southern pylon (opera house side) that tells all about the construction of the harbour bridge. The ticket also gives you access to the lookout at the top of the pylon. Well worth the entry price alone. You could also book online before you leave and trek up the arches of the Bridge with BridgeClimb.
5. If you have the time, take a wander through Darling Harbour. There is always something happening there, day or night. At the very least you can catch some night life at the cockle bay wharf bars or visit the Aquarium or Aussie wildlife places down there that won't take all your time.
6. If you do have the time, take a ferry to Taronga Zoo. Entry is expensive, but its the one place where you can see the most unique Australian animals all in one place.
7. If you are into Greenery and beautiful plant life, you can visit the Botanical Gardens which run a good length of eastern edge of the city centre. There are several gates into it, with one right next to the Opera House. Reminds me, you can also take the 10 dollar guided tour (prob more now) of the Opera house. Buy your ticket from the lower shopping level on the western side of the Opera House forecourt.