I'd head in to town and start exploring. Get a map of SYD at the tourist info booth at the airport.
Head over to The Rocks and Circular Quay. At this location you can see the Sydney Opera House and the Sydney Harbor bridge. Have something to eat in this area and relax as you overlook the harbor. The Rocks has some cute shops.
Depending on your time, I'd take a ferry from Circular Quay A$12 return to Manley Beach and walk around. The 30 minute trip provides awesome views of Sydney Harbor, the bridge, and the Opera. The ferry runs every 30 minutes. AT Manley, you can walk a few blocks to the beach, where many restaurants are located.
Sydney has a great park in the center of town called Hyde Park. Nice place to spend some time.
Meet up with local Flyertalkers and have dinner/drinks. ^
There is a shuttle into town from the airport A$9 1 way. The train is A$10.
If you read the Australia forum, you may find additional info.
Dan