My foreign friends all enjoy Darling Harbour - myself as a Sydneysider, I can't see the attraction but visitors seem to love it. There's a great aquarium there, pricey but worth it.
I'd take the ferry to Taronga Zoo and spend at least half a day there. The ferry takes you to the bottom of the zoo, you can then take a cable car to the top with a great view of a lot of the animals, then walk your way back down to the ferry for your return. It's a fantastic zoo, you can get up close to the koalas and walk amongst the kangaroos and wallabies, and there's a really good platypus exhibit - you're unlikely to see any of these animals in the wild. The ferry trip is great, it's closer than Manly.
Katoomba in the Blue Mountains is good too, a two hour train trip from Sydney. A meal at the Paragon is good (for the ambience, not so much for the food) and a visit to the cocktail bar at the Carrington is a real experience. The weather can be dodgy, very cold in winter, and if it's been raining the walks are risky (it gets very slippery). I'd recommend two nights in Katoomba so you get a full day there, with an early start to get the best of the Three Sisters, walk down the Giant Stairway and head to Scenic World (there's a funicular that can take you back to the top, you'll be stuffed after the walk).
The Carrington
http://www.thecarrington.com.au/ has a good restaurant, and Bistro Niagara just opposite the railway station is great too - it would be smart to book in advance if either of these appeal to you.
Enjoy your stay in Sydney, it would be good to hear a report back from you when your trip is over.