I stayed at the Holiday Inn in Darlinghurst...if you walk out of the hotel to the left, keep looking to the left for a stairway, it will take you down toward the wharf, and then you can walk through the park and botanic gardens to the Opera House...a great walk, and a great approach to the city and the Opera House itself. There are good maps once you get to the park, and the walk along the water, to the point where Mrs. Macquarie's chair is located offers great views, and photo ops.
Walk out the hotel to the right, up darlinghurst road and go have breakfast, or dinner at Una's on victoria street. Again, out the hotel door down the street to the left, but on the right side of the road is a great thai take out place, good cheap food (the name escapes me but hotel staff can tell you).
The hotel is right in the King's Cross area, there are abundant very cheap internet cafes (interspersed between strip joints) and it's a wild scene in the evenings. The hotel is also, literally, above the King's Cross subway stop, the blue line, very convenient for getting to/from downtown...and the walk is also worth it, a good stroll from city center along William Street and walking pretty much anywhere is safe, at least I felt so, even at night (as a female travelling solo).
I endorse others' recommendations about ferry rides. Bottom line is, don't pay for the touristy boat trips when you can take a ferry to Manly or Watson Bay for much less and get the same experience.
Have a great time...it's an AWESOME city to visit!!!