Long-term Sydneysider here (until three years ago):
I see you’re from Southern California, so you may have had your fill of excellent zoos and stuff, but if you’re not morally opposed to such attractions then you should definitely check out Taronga Zoo and the Sydney Aquarium. The former has a good collection, and is located spectacularly by Sydney Harbour. The latter is fantastic if you’re a lover of sealife; the large tanks where you walk underneath the sharks and rays are amazing.
It’s good that you’re getting out to see the Blue Mountains. The attraction that Thaidog was referrring to is the Three Sisters, not the Three Witches :). It’s located right near the town of Katoomba, and is really worth a look. Another great view not far from Katoomba is Govett’s Leap. Try to go on a day without fog, however, because if the weather is bad you can barely see the rock formations. If you’re already up in the mountains and have a few more hours, you might want to check out Jenolan Caves also.
I’m not sure of your exact itinerary, but if you’re in Sydney on July 6 and you’re a sports fan, you might like to take in a game of Australian Rules football. The Sydney Swans are taking on the West Coast Eagles at 1:10pm that day, at the Sydney Cricket Ground. You could also catch any one of a half-dozen or so National Rugby League games.
If you like good seafood, i second Big Nik’s suggestion of lunch at Doyle’s. A good way to get over there is to take a water taxi across from Circular Quay. It’s not a really long ride, but it’s fun and gives you a good look at the harbour and the foreshores from the water. For more leisurely water transportation, take a Sydney ferry over to Manly. It’s nice to sit outside on the deck of the boat, but it might be a bit cool in July for this.
The Queen Victoria Building is lovely and definitely worth seeing, although the shops inside are generally horrendously overpriced. I would also suggest a visit to the Sydney Town Hall, which is next door to the QVB, but their website says that the building will be closed from July 1 for renovations. I’d probably avoid Kings Cross, mainly because it’s just rather seedy and boring. Neither it nor Redfern are as dangerous as some people seem to believe. Most of the hype surrounding these areas is generated by people who:
a) have barely (if ever) been there
b) are more comfortable in the suburbs than in the city
Anyone with experience in a big American city will find Australians’ notions of a “dangerous part of town” pretty laughable. That’s not to say that nothing bad ever happens; only that the hype usually far exceeds the reality (and this is the case in many US cities too).
I find The Rocks a bit touristy, but there are some good pubs in the area, including a few that make their own excellent brews. Some of these are worth checking out, especially on a quiet afternoon. If you don’t like waiting ages to get served, avoid the area on Friday and Saturday nights, as it gets very busy.
One thing i never got around to doing before i left Sydney is climbing the Sydney Harbour Bridge. This costs $A145-175, depending on what time of day you go. It’s not really cheap, but if you get a nice day the view from the top should be well worth it.
By the way, for those with high speed internet access, the Sydney Opera House has a fantastic virtual tour, in which you can “walk” all around the building, inside and out.