Sydney is amazing and I'm going back this Christmas!
Anyways, Since it's summer, I would definitely go to:
Bondi - Great Beaches, great food, and a great mall (Westfield Bondi) right near it, with shops ranging from great Australian chains such as Witchery, and Country Road to Higher end stores such as Versace. A bit for every body =)
Paddington - Oxford St is the place to be - lots of cool bars, and clubs, and GREAT shopping. There are vintage stores mixed in with flagship boutiques for Australian brands such as Sass&Bide, Kirrily Johnston, Scanlan & Theodore and Alannah Hill.
Pitt St. Mall & Strand Arcade & Castlereagh St - Check out Pitt St Mall for the two biggest department stores in Australia. Myer is kind of like Macy's whereas David Jones would be comparable to Saks. Along Pitt St. Mall, there are also Australian chains along with Ralph Lauren etc.
Strand contains higher end stores such as Third Millenium, Alannah Hill, Zimmermann. Beautiful dresses to buy!
Castlereagh St is our 5th Ave of NY. Here you'll find Hermes, LV,
Gucci, Ferragamo, and Chanel. Alongside Martin Place - where Castlereagh st leads to - and where the Westin Hotel is located (in my memory), are amazing stores such as Evelyn Miles - who are actually having a sale right now! Evelyn Miles stocks designer shoes such as Lacroix, Manolos, and Choos.
Anyways.. this is where I'd hit up for shopping. Check out the australian vogue forums for info on the Australian sales etc. at forums.vogue.com.au
As for places to eat, it depends what you're looking for. Right now, Sydney probably has more Asian food (authentic too!) than western (such as italian) food, more healthy food than fast food. Sydney is a very heatlh conscious city - so definitely don't go to Australia for pizza or any other American takeout food - our dominos slices are about 1/2 the size of a normal American slice (although there are a lot of good woodfired pizza joints). Gelato is popular, and so are Juice bars. Try them - they're everywhere. Also, sit down at a cafe and enjoy a meal =) In Australia - you can eat anything in a cafe - from sirloin, to meat pie, to a mortal sin cake.
Hope you have fun!