newbie, if you love beautiful ocean scenery, you have got to check out The Great Ocean Road, it's this 300 km stretch of amazing coastline just south of Melbourne, about an hour's drive!!! Unbelievable rock/cliff formations set dramatically against the raging sea... The Twelve Apostles was unreal!!!
And definitely go to Frazer Island and Whitsundays, up north. Gorgeous white sand beaches and lakes. I wish I had time to check out Tasmania.
As for regular things to do, I loved walking around Sydney's Botanical Gardens along a scenic path that took you straight to the Opera House. Do it near sunset, it's lovely.
Also, my favorite spot in Sydney was Manly Beach, a short ferry ride from the heart of Sydney...stay out there for a few days if you can. It's a cozy beach town and everyone's super friendly, lotsa surfers, fun bars, and for $100 you can dive with the sharks and rays at the local aquarium.
Touristy attractions like the Sydney Aquarium had one of the largest shark collections I'd ever seen and it was awesome. If you stop by a local hostel, like Wake Up Sydney near the train station, you get a coupon for 15% percent and pay $18 instead of $30.