As suggested Powerhouse in Sydney: http://www.powerhousemuseum.com/
I’d definitely recommend stopping in Canberra. Lots of people bag it as boring, but it is a stunning planned city (by Walter Burley Griffin) and it has my favourite museum in the world, the Australian War Memorial:
http://www.awm.gov.au/.
This is the view from the War Memorial across to Parliament house:
Also, drive around Yarralumla, all the embassies are in that suburb in between suburban houses. Great to see these monuments to National pride and ego, full of flags and security guards and fleets of limos, next door to a house where a guy is mowing his lawn with his shirt off.
Melbourne is great too, you can get the “Melbourne Design Guide” as a walking/ tourist guide. Melbourne is very easy to get around with public transport: Home - thedesignguide
Over Christmas/ New Year/ January you can see some cricket at the MCG (the Ashes are about to start) with 100,000 of your closest friends or the Australian Open tennis.
The drive from Melbourne to Adelaide via the Great Ocean Road is fantastic. I’m biased to Adelaide but it’s about the quarter the size of Melbourne, but with way better beaches.
The Art Gallery of South Australia has a great William Morris collection Art Gallery of South Australia - AGSA plus the original Lockheed lounge, purchased from Marc Newson’s first student exhibit of it:
Adelaide’s got so many wineries, very nice drives from cellar door to cellar door, but you can do bus tours so you can drink and not worry about driving. I’d recommend Coriole in McLaren Vale , Sevenhill or Annie’s Lane in the Clare Valley , Bird in Hand in the Adelaide Hills , and Penfolds in the Barossa, but really anywhere you can’t go wrong.
Adelaide in January has the cricket (Adelaide Oval redevelopment will be finished), the Tour Down Under (http://www.tourdownunder.com.au/ and in Feb/ March if you are still around the Festival and Fringe (http://www.adelaidefestival.com.au/, http://www.adelaidefringe.com.au/) plus the Clipsal V8 race (http://www.clipsal500.com.au/).