Cheap drinks? Your in Sydney so thats a hard one. Go to a small local pub away from tourist areas for that. Or for people watching, fish and chips, nice pub with great view back to the harbour bridge go to Watsons bay from there you can walk along the beach to some nice little coves and beaches.
Other than that the usual tourist/local spots are the rocks area, (some nice old pubs there) Bondi Beach-go to the surf club bar overlooking the beach at the south end. Cheapish drinks and one of the best views in Sydney-continue the beach walk around to Bronte beach or further-one of the best things to do in Sydney.
Kings Cross is the most overrated area in Sydney IMHO, some great food ,bars but be careful walking around at night it can have a bad vibe at times. Sydney is such a touristy city, locals mix with tourists everywhere so its hard to tap into a "local" scene as such.