I just booked a couple of nights at The Menzies for my mum and dad... surprisingly cheap (I think it was about $170 per night, but that was for early next year, so may have been an advance purchase special or something like that) and it's just across the park from Wynard train station which is on the City Circle loop, so it's easy to get to Circular Quay for the ferries if you wanted to go over to the zoo.
My kids really loved the Sydney Tower Eye (Centrepoint) when we went to Syd for the day a couple of months back, but if you just rock up on the day it is pretty expensive - it was about $80 for the 5 of us to get in. If you book online and you pick a non-peak period it is quite a bit cheaper - I think it was about $47 if we booked online and got there before 10am. They also really enjoyed Powerhouse and if we had time, we wouild have gone to the Maritime Museum at Darling Harbour as well, but we only caught the train down for the day, so we ran out of time..
ETA: Here's the network map for Cityail
... if you are looking for accommodation, try looking mainly at places on the City Circle loop (Central / Town Hall / Wynyard / Circular Quay / St James / Museum) - the trains run pretty much continuously and go to most of the touristy type areas.
This post has been edited by ~Nic~: 09/11/2012, 09:53 AM