Oddly, Sydney doesn't really appeal to me much.
The foreshore area near the Sydney Harbour Bridge is worth wandering around, along with the Opera House. Even the bridge has commercial tours over the top of it.
The scenery all around Sydney Harbour is pretty grand to see on the ferry services or from cliffs at the heads. The Hunter Valley would be an obvious choice for those who like wine. To the west is the Blue Mountains which has a decent history and plenty of hike opportunities.
Of course, Sydney has a pretty high participation rate in the harbour yacht racing, and what's out there might be interesting. My advice is to stay away from Kings Cross, but you may have a more tolerable view of just about every type of human diversion that exists.
Apart from that, Sydney in my opinion is not good for cycling.
Sorry to sound negative about it, but I think there are much better elsewhere places in Australia to visit.
And almost without a doubt, your wife will have had a great time in New Zealand.