August is one of the best times to go to the reef, in summer it's the wet season and terribly humid and not very pleasant. Cairns is hot all year, even in winter. You will need shorts and t-shirts, sunglasses, hat. Don't miss the opportunity to visit Cairns in August.
Sydney will be cool to mild, coldish at night. You will need a couple of jumpers, jeans etc. There may be some quite warm days but it isn't likely to be shorts weather. Adelaide will be similar, usually a couple of degrees cooler. Allow for some rain, Sydney actually has more annual rainfall than London.
Auckland will be quite cold. Is there some reason you will spend so long there? It's not the most interesting part of NZ.
You might also like to include the Gold Coast for a few days if you can, your nine year old will love it and the weather will be pretty good.
In Cairns as well as the reef you can visit the Daintree Rainforest for a wildlife spotting tour, see crocodiles in the wild on the Daintree River, and take the Skyrail or the scenic railway to Kuranda where there are several things to see and do.
In Sydney you can visit the Rocks historic convict built area, take a ferry on the harbour to Manly beach, do a harbour bridge climb, visit the aquarium, take a day trip to the Blue Mountains (it will be cold) where you can ride the world's steepest train up the side of the mountain and ride a cable car as well as going for a bushwalk, do an evening harbour cruise, walk around Darling Harbour, visit Taronga Zoo, take a guided tour of Fort Dension which is an interesting historic fort on an island in the harbour. Visit artsy Newtown, 'little Italy' Leichardt, and 'little Vietnam' Cabramatta.
In Adelaide if you make it you should definitely visit Warrawong wildlife sanctuary for an evening guided tour - this is far better than any other wildlife park in Australia. There aren't too many interesting tourist activities in Adelaide itself (it's a great place to live, but not much of a tourist destination), but there are plenty of great places to visit if you can take day trips/short trips from Adelaide, such as the Barossa Valley wine region, Kangaroo Island, and the Flinders Ranges.
Auckland I won't comment on, I'll leave that for a Kiwi.
The Gold Coast is Australia's main family tourist destination, and has great beaches, theme parks, and lots of tourist activities perfect for kids. You could stop there on the way from Sydney to Cairns.