There are two different ways to climb the bridge, and the more difficult one to find out about is by far the cheaper. Sydney has a lot of Aboriginal heritage, so you can go on a tour of that. The Blue Mountains are a world heritage site, and you should be able to get there without too much expense - there is a train, and the Three Sisters are not far from the station, or you can get a day tour there. Of course, you could go to world famous Bondi Beach, or out to the harbour heads by bus. Go on a trip on the ferry, walk around the harbour from circular quay, visit The Rocks, go to Cockatoo Island by ferry (it has a very interesting history, and is used for alternate art installations etc.).