If you want authentic vietnamiese food go to Flemington - there's this really dodgy looking place that always has a massive line.
If you want authentic Shanghainese food go to Ashfield - there are plenty of nice places there.
If you want authentic Korean food go to Strathfield - little Korea awaits.
If you want authentic Italian go to Leichhardt.
blahblahblah Japanese food is everywhere in the city if that's your thing.
Used to be the thang but Adriano Zumbo has great/wacky macarons.
DEFINITELY GO TO THE ZOO AND AQUARIUM!!!!!!!!!! THIS IS A MUST!!!!!!!!!!! Also if you only intend to hang around the city, I wouldn't really bother with hiring cars or stuff, the public transport is sufficient and there's family fun day on Sunday where you pay $2 for your whole family to travel the whole day Also there are free buses for tourists that go around the city... you should go to Cockatoo Island if there are any events there by ferry... usually pretty fun. Idk if you want fresh seafood but there's the Sydney Fish Markets... again that's kinda crowded with tourists.
If you're into art, go around Newtown, Surry Hills (Newtown also has awesome food like PIE TIN!!) and the city - MCA. If your kid was here and a bit older there was an Anish Kapoor exhibition. Hmm if you like looking at buildings I can go further into this ^_^ Usyd is pretty nearby, but there's only one scenic side (the other side is fugly as)... Central Park has a new botanical building... UTS has this weird porous building... and yeahh.......there might be some light shows as well... not sure when this is... but we recently had Vivid.
Edit: Opera Bar is kinda expensive but a nice classy place on the opera house for drinks. Lol i pretty much reiterated what's already been said... Idk... all I do in Sydney is eat nice food. IMO some of the places have food that tastes even better than their original place ha. ha. ha. ha... Hyde Park is quite nice to walk around, there are also sometimes sculpture exhibitions. Dang it you miss Sculpture by the Sea It's a really pretty time to go to Bondi.
Darling Harbour also has a pretty awesome playground hee
I think Coogee is a bit better than Bondi and less crowded... I guess get fish 'n' chips there?