Whip, head down to Utopia. Its on George Street, down opposite all the movie theatres.
http://www.utopia.com.au/
Also, there is a huuuuge Cathedral, and an Art Gallery in The Domain. And near there is an awesome museum.
The Rocks is quite a pleasant place to go. Manly is easy enough to get to and potter around. Bondi is amazing, definately make tracks to there. I've never seen a beach quite like it.
Theres also that abstract gallery with the 2 huge match sticks out the front... can't remember where that is located though.
These may seem quite touristy.. but they are the kinda things I did when I lived there, besides egging people's houses, having lans, and learning to skateboard over in East Ryde. You're welcome to try those things too =)