What everyone else said (except Queensland is too hot!).
Taronga does Roar N Snore where you go glamping for a night smack bang in the middle of the grounds with the best views of Sydney Harbour. As soon as my girls are old enough I'll be making it an annual family event.
Janolan Caves in the Blue Mountains are wonderful if you're into nature stuff. The cable car ride through the mountains is breathtaking.
Manly Ferry from Circular Quay - you can do a shark dive at Manly Sea Life.
Canoe from (can't remember where) up towards The Heads.
The Opera House has some lovely bars out on the forecourt - there is opera-y stuff going on inside too.
Pub crawl through The Rocks.
The Sydney Festival starts in the first few days of Jan - lots of arty farty and foodie stuff going on.
OpenAir Cinema for a unique movie experience - don't sit under the Moreton Bay Figs or you WILL get bat shit on you! I can hook you up with tickets as they sell out pretty quickly ( my BFF helps run the event).
Panning for gold and Featherdale Wildlife park - good ones for kids.
There's always some football thing on (Rugby League or Rugby Union) - come see how the game was supposed to be played. Avoid the cricket - more boring than batshit.
You can do go karting at Eastern Creek - so much fun!
Try to see a sunrise over the beach - being an east coaster seeing the sunset over the beach on the West Coast of the US was an amazing experience for me.
I'd look at AirBNB for accommodation. I always recommend people stay in Balmain/Rozelle for the food, bars and villagey feel (a few Ks from my 'hood), Manly for beach life, Maroubra/Coogee for southern beach gangster/backpacker life or the Eastern Suburbs if you're soulless.
Our CBD is a bit of a ghost town but for shopping/nightlife.
And a G2G with me!
I may be biased having lived within the same 10k for virtually my entire life but Sydney is a beautiful place.
[This message edited by SBB at 7:32 PM, January 2nd (Thursday)]