Sydney is a large city so has many of the same attractions as many other big cities but with the added extra of the opera house, decent beaches, loads of bars, clubs, live music, nightclubs etc. Don't worry there is plenty to keep you busy!
The blue mountains are a couple of hours on the train out of Sydney but it is beautiful and I would definitely recommend it!! The trains there are regular and cheap and there are a couple of decent hostels there as well as rooms above pubs etc.
Believe me you will soon get into the swing of things and you definitely won't be bored!!