Sydney is a fantastic city with lots to do and see. Six days will really allow you to settle in, see the sights and experience our 'culture' and even do a day trip out of the city.
You will need a jumper and jeans in August - but I don't consider that a reason not to come. The only thing I'd say is that you wouldn't be tempted to go for a swim at Bondi Beach! So if you wanted to experience Sydney's beach culture, I would say choose another time to visit.
Below are some great places to visit in Sydney. I have also included some locations off the beaten tourist track. Sydney CBD is best seen by foot, if the weather is inclement, you can take a free shuttle bus or the trains.
* Circular Quay is the tourist hotspot - from here you can walk around to the Opera House, stop in for a drink at Opera Bar and continue around the harbour's edge through the Botanical Gardens, Mrs Macquarie's Chair (great views looking back onto Opera House and Sydney Harbour Bridge), Garden Island, through the Domain and up to our grand church, St Mary's and through Hyde Park where you will end up in the centre of Sydney's CB - you can visit the big shops and the QVB building. Another lovely place to stop is at Andrew Boy Charlton swimming pool cafe - great good and soak in the views. Depending on your pace, I'd say that walk would take about an hour and a half. I call it the 'best of Sydney' walk.
* From Circular Quay, catch a ferry over the Taronga Zoo - the animals have the BEST view in Sydney and you get a different perspective from the zoo. The bird show is great - don't miss it. Best to plan for a full day at the Zoo, you may wish to take a walk up to nearby Mosman from the zoo. The houses along this strip are amazing and worth millions of dollars. Mosman also has many cute boutique shops. Full day.
* From Circular Quay, catch the ferry to Manly. Sydney is over the other side of Sydney heads, so the scenery is great! You wouldn't want to go swimming at the beach in August but you could plan to have breakfast or lunch over that way.
* Take a bus to Bondi Beach. Every tourist must see Bondi beach but I don't recommend hanging around the area too long - I would continue to the South of Bondi and pick up the coastal walking track from Bondi to Coogee -it's about 6km and trust be you wont regret it. Its absolutely fabulous - I never get sick of that walk. http://bonditocoogeewalk.com.au/
* I don't recommend going to Sydney Tower - I consider it to be a rip off and I think the views are best at the cocktail bar at the top of the Shangri-La Hotel. Sure, you might pay approximately $20 per cocktail, but the views are absolutely spectacular. Best to go in the evening and you can see the city lights!
* From Circular Quay take a walk under the Harbour Bridge and continue to the finger wharves at Walsh Bay - there are some fantastic restaurants in this area so you are sure to find a nice place to eat. From Circular Quay it may take you 20-30 mins to get to Walsh Bay.
* The Rocks - explore the Rocks - where Sydney all began. Have a drink at the famous Lord Nelson pub or lunch/dinner at the resturant Fish at the Rocks. Don't miss Observatory Hill, it's just beautiful up there with magical views. Time your visit for the sunset. It's beautiful. There is also a cafe in the grounds.
* Personally, I would consider a Sydney Seaplane ride over walking the Harbour Bridge. I have done both and consider that there is better value for money in the seaplane experience. You can really understand Sydney from the sky. However, having said that, you do get a great history lesson of Sydney and in particular the construction of the Sydney Harbour Bridge on the walk so please yourself.
* China town is okay - paddies market is great to pick up some Sydney trinckets and there is some great cheap food around. I wouldn't eat along the touristy Dixon St - venture a little further and the prices will drop and food becomes better (IMHO).
* Likewise Darling Harbour is a tourist hotspot and you should check it out. Cockle Bay is a great place to have a drink. Sydneysiders don't spend a lot of time around Darling Habour although a lot of money and effort has gone into sprucing the place up with some great restaurants. I need to go back to give it another go. You can incorporate Pyrmont and the Fish Markets from here.
* Blue Mountains - given that you have six days, you can afford to go to the Blue Mountains. It will be significantly colder here so you should wear appropriate clothing. There are so many tours you can take from Sydney which will take you to Katoomba and the Three Sisters lookout etc.
* Hunter Valley - another day trip option is to go to the Hunter Valley, Sydney's closest wine region. Great wines and cheeses. Only 2.5 hours from Sydney so definitely manageable for a day trip. Beautiful scenery too.
I have many more suggestions - if you'd like more just let me know. If you like good resturants, I can also give you some suggestions - I like eating good quality, fresh food which is not overly expensive.
I think 2 days at the snow is not long enough. From Sydney, it will take you 5.5-6 hours to get to Thredbo or Perisher and same on the way back. You need at least 3 days, but even then, alot of your time will be travelling. Plus the snow is expensive so I consider that you really need to get the best value for your money. Personally, I'd give it a miss.
Hope this helps.