If your stay includes a Saturday night there will be fireworks at Darling Harbour. There are lots of places to eat around there and you could then walk along the water through Barangaroo to Circular Quay (or do it in reverse). If the weather isn’t great there is an iMax cinema there.
Get the ferry from Circular Quay to Manly. Nice beaches and good atmosphere.
The Botanic Gardens are great for a picnic and you can combine it with seeing the Opera House.
Go see the Christmas Tree in Martin Place, the Christmas windows in David Jones on Elizabeth Street and the Christmas lights projection on St. Mary’s Cathedral.
Luna Park would be fun for the kids (and you have lovely views).
Boxing Day lunch cruise is a good idea, look at Scoopon for deals.
Bridge Climb is amazing but $$$$$
If you don’t have an ethical objection to zoos Taronga is one of the best.
I think the Fish Markets at Pyrmont will still be in their current home for Christmas (they are due to move). Good for a wander and the best fish and chips, just beware the seagulls.
Basically Sydney is incredible around Christmas as long as the weather plays nice. The wow factor is big Aussie blue skies and loads of things happening outdoors (check out Time Out for listings of what’s on). A rainy Summer is disappointing but there is nothing you can do about that.