I took a 2 month trip through Australia a couple years ago...
My best advise in Sydney( and everywhere else) is to stay in a hostel. They're actually pretty nice and you can get private rooms for less than a hotel room still. plus you get bars usually in them and other younger people to meet and talk to.
Just make sure you book ahead for your space... my friend and I ended up having to stay a couple nights in an all illegal asian immigrant stink hole one night because it was the only thing left... they were very nice and I learn some vietamese but still I wouldn't recommend it.
There is also a Spanish restaurant in sydney's downtown. I think it's on Pitt Street??? I can't remember what it's called but if a girl could have wet dreams this would be in mine. You must eat there. Must.
Also clearly go see the opera house and the bridge...
there's some old neighbourhoods that are worth a stroll, as well as a harbour with a navy museum and an underwater world with huge sharks that you can walk underneath in a glass tunnel...
The sydney tower is fun, try going up at sunset it's pretty amazing.
Bondi beach is huge and packed... but worth a trip. Also a lot of people in australia sun bath naked... not just the hot chicks though sorry- old gross men too.
I'll let you know if I think of any more, or you can pm me if you'd like more or some pics
