winglovesall said:
Hire a car out to the Blue Mountains. It takes 2 hours approximately to get there.
Walk around the city. Catch the Blue and Red Explorer buses that can tour you around major attractions of Sydney including Bondi Beach, Sydney Opera House, Manly .... .
Of course, the Mountains! I used to live in Blackheath until 2003. Beautiful place. Get a train up to Katoomba or Blackheath. Katoomba has some great lookouts (you'll need a taxi to get there if you don't have your own car) - the Three Sisters is world famous and Echo Point is beautiful.
Blackheath has some nice lookouts as well - Govett's Leap, down the end of Govetts Leap Road (it's the main road of Blackheath - turn right at the lights coming from Sydney, or else cross the railway, then the highway, then turn right and walk up to the end of the block, and then turn left - taxis etc. are there). It's got a couple of walks and so on, which are a great way to have fun without spending a cent.
Evan's Lookout, which is down the end of Evans Lookout Road - that's the first main road coming in from Sydney by car.
Hat Hill Road leads to Hat Hill (can you guess what they name the roads after in Blackheath?), which is a good climb and has an okay view. You can keep going down Hat Hill Road (it's about 2km then 7km more of dirt road) and there's Pulpit Rock, which is great. There's a walk that takes about two or so hours from Govett's Leap to Pulpit Rock (I think - one down the end of Hat Hill Road anyway) - it's great fun but is not a loop so you might get stuck with no car. (I used to live on Hat Hill Road - well, actually, St. Elmo Ave., a tiny road parallel to Hat Hill right down near the end.)
There are others in Blackheath as well, though I can't remember exactly how to get there. Going left at the lights coming from Sydney leads to some really nice place, eventually.
"That thing that falls down the cliff face" that Bevlina mentioned is the Scenic Railway - good fun. There's also the Scenic Skyway there, which I've never been on. There are some good walks and scenery around there. The Scenic Railway is the steepest railway in the world - once a mining railway, now a very famous tourist attraction.
And of course there are plenty of other places as well - they're just the ones off the top of my head!