Funnily enough Sydney is my favourite holiday destination in the whole world.
The place that I stay at every year is the Lane Cove River Tourist Park. Although not exactly IN THE CITY, it is only a 10km drive away and is accessible directly by bus along Epping Rd. Another option to get to the city is to take the bus to Chatswood and then the train. The buses are walking distance from the Park.
The place is great. There is a TV and Internet Room (internet costs $2 per 20 minutes), barbeques (free), camp kitchen (fridges, microwaves, hot water system etc - free) and 2 swimming pools. Tent sites cost $28 and $32 (unpowered) off peak and $32 and $36 (powered) during the peak period.
If you're an animal lover you will find heaps of animals there - lizards, bandikoots, lorikeets, kookaburras and rabbits. The National Park is right next door and is free.