Sydney is lovely, but definitely plan to get out and about. It's not like London where you can spend days exploring. Must sees are the Harbour (don't pay for a boat tour, just use the local transport ferries), Luna Park, Opera House, Australian Museum, Museum of Sydney, Art Gallery of NSW and the Botanic Gardens. Southern suburbs are nice, the Coogee to Bondi coastal walk is lovely. Ferry to the Northern Beaches for good coffee and rich hipster scene. Marrickville (Inner West) for the slightly less rich hipster scene, and to see what non-beach Sydney is like.
Blue Mountains as others have said are stunning. Maybe plan an 'adventure' activity rather than just the genric coach tour type thing, as the real beauty is away from the roads and towns - canyoning perhaps?
A few days trip down the south coast would be really nice. You can train as far as Nowra, but I would recommend hiring a car and taking four-seven days to do the following trip:
Royal National Park (just south of Sydney) for an intro to the Australian bush. Stop off at Symbio Wildlife Park (not as big as Taronga in Sydney Harbour but much quieter. It's illegal to cuddle koalas in NSW, but you can get pretty close, and they have all sorts of awesome other Ozzie creatures). Then through the northern suburbs of Wollongong (Scarborough to Thirroul) for some beautiful scenery and gorgeous beaches and rock swimming pools. Wollongong itself is a bit of a post-industrial city, but has pretty beaches - good to stop off for a surf lesson? There is a stunning cycle path all the way up the coast from North Wollongong to Thirroul - you can hire bikes from North Wollongong and have a beautiful trip taking in lots of beaches and cool cafes etc. Then back in the car and down to Shellharbour - awesome easily acessible snorkelling at Bushranger Bay. Then Kiama - nice little town and cool blowholes. Then Seven Mile Beach (self-explanatory, beautiful!), and on to Jervis Bay (as others have recommended). Jervis Bay is stunning - white sand beaches, whale watching, rock climbing. If you are a camper, I would 100% recommend picking up some cheap camping gear while you're over there and camping at the state campgrounds at Green Patch and Honeymoon Bay in Jervis Bay - campfires with the possums, share your brekkie with the kookaburas, and watch sunset on the beach with the kangaroos.
Have fun!