Hiya
Aside from hiring a 'limo'/shuttle with carseats fitted, you also have the option of taxi's. In Sydney, it is compulsory for kids to sit in carseats/boosters, so a select number of taxi's have them on board. When standing at the taxi rank at Sydney airport, the attendant on duty (or sometimes other drivers if no one is present) will pull you to the side and wait until a taxi with a carseat comes by (normally a station wagon). I am not certain however about baby capsules (and in that case you may need to go through a limo/shuttle service). If your bub can sit upright ok in a carseat though, you should be fine.
I would recommend Taronga Zoo to those with kids walking, and especially for toddlers. However, make sure you pick up one of the 'Sydney' tourist guide books from the airport before you come through customs as it should have some handy discount coupons for various attractions, as they can add up quickly (and Taronga Zoo is quite expensive)! Taronga Zoo has great photo opportunities back to the city, however it is on a hill, so don't go if it is windy, as the Gondola that takes you up and down inside the zoo doesn't run when it's windy. The bird show at the zoo can be pretty spectacular if you want to see some of Australia's birds 'in action'.
I can also recommend the Sydney Aquarium as an interesting place to get up close and personal with fish, sharks, seals and a quite spectacular coral reef, including 'Nemo' tanks. They also have 'touch tanks' for the kids.
If reptiles are your thing, you might want to see if you can hire a car and go see the 'Gosford Reptile Park'. It's got some great snake/croc/gator displays, and the kids can feed kangaroos. There are a good range of Australia animals on display, you can have your photo with a snake around your neck, or cuddling a koala.. and the funnelweb demonstration (even if you're arachnophobic) is pretty cool.
The Australian Museum is also pretty good (as someone suggested)... and if your little one wants to do something extra special there, for $5 per child (for the first 30 children on select mornings) they run a dinosaur experience for the under 5's, where they get to wear a dinosaur tail and stomp around the museum with a guide who shows them through the dinosaur wing of the museum.
I'd also strongly suggest a harbour cruise... Sydney harbour is lovely by day and especially by night (although that isn't necessarily easy to do with a child).
If you're staying in the CBD, go down to Circular Quay and take a ferry to the zoo or Darling Harbour (where the Sydney Aquarium is). Even those short ferry rides are fun for the kids.
There is a free park (not fabulous, but ok) in the Darling Harbour Precinct, near the IMAX theatre, and if you go on weekends, there is often a merry-go-round running.
In winter, the beaches aren't all that great, but Manly is ok (another ferry ride and some good food). I've never seen what people see in Bondi, but Coogee which is not far away is nicer.. however parking in that part of Sydney can be difficult.
The "must-see's"... Harbour Bridge, Opera House, The Rocks, and botanical gardens are all within walking distance of Circular Quay.
I don't know much about the hotels and motels around the city I'm afraid.. but I do know where most of the big ones are located if you want an idea of proximity to stuff.
If you need more info on Sydney, feel free to send me a priv msg.
