are you chinese? star city is the only stop you need my friend!
alright...i'm being stereotypical
public transport is pretty shit...i mean very shit! i rather be stuck in traffic in my car for over 1hr than take a 30min train ride.
i've was born and raised in sydney so here a some places:
in the cbd you have all the typical landmarks such as Harbour Bridge which you can climb. there is the Opera House. if you really want to know how sydney all started you can take a historical tour around The Rocks. For some scenery, take a stroll through The Botanical Gardens. Main shopping departments stores are David Jones and Myers. Both very close to each other. If you like fish, take a walk through Sydney Aquarium and then walk along Darling Harbour. If you don't like fish you can still take a walk through Darling Harbour. For the beaches, Bondi Beach or Manly is very popular but there are others too. I suggest going to Cicular Quay and take a ferry ride across the harbour and stop at Manly. Walk along the Corso and go right to the end near the beach and get some fish and chips for lunch. The family should enjoy that. The red light district is Kings Cross but its not what it used to be. Still there a street walkers around and plenty strip joints.
out of the CBD, you can goto the Royal National Park. This is where you need to hire a car. You can take a walk through the bush and cool down by the beach and have a picnic. You can visit Olympic Park but pretty ordinary when nothing is on. theres not much to do outside the cbd unless you like eating
for some good asian meals you can head out to the subrubs of Ashfield, Newtown, Chatswood.
hopefully i've given you some useful suggestions to try. all suggestions are family friendly except maybe Kings X
i've been to melbourne and its a fantastic city. i would drop sydney in a heartbeat. i only go down there for the F1 so i've always stuck around the CBD. even though i haven't seen much except for the 12 Apostles and driving around Great Ocean Road so i can't give much info.
I've been to Brisbane but i rather stick with Gold Coast. Has much of a fun atmosphere and there are theme parks to take the kids. If you get multple passes they come a bit cheaper.
IF you really love nature, either Darwin or Cairns is your best bet. In Cairns you can see a small part of the Great Barrier Reef. In Darwin you can visit Kakadu national park, cruise though Katherine Gorge etc. If you goto Darwin, you must try fishing.