I have only been living in Sydney since January but can definitely agree that the trains here are great except in peak hour times! You might have trouble getting your pram on at those times, DH tells me he can't even get on the train sometimes as its too full and he is often crammed on standing. Also I have only taken my baby on the train once so far (he is still pretty new) but the lifts sometimes are out at various stations (e.g. currently they are doing maintenance on the lifts at the Town Hall station so don't know if they are in use there) so it can get tricky if you have a pram. But that aside they do have space for prams like puddleduck mentions and there is a train that goes via the airport which is about $15 to the city if you don't want to catch a taxi to the city but you can ask for a taxi with a car seat though not sure how lucky you will get! I do love the frequency of the trains and have never had to worry about looking up schedules
Oh and just a note on carseats, here it is the law to have them anchored into the car (I assume taxis will have anchor points installed), we didn't know this and had been given a NZ car capsule for our baby which meets NZ safety standards (and doesn't need to be anchored) but not Aussie ones.. so we will have to get one when we finally buy a car.
If you pay $20 you can get a day unlimited transport pass on all buses, ferries, and trains. You could go over to Manly beach on the ferry and also ferry between circular quay and darling harbour to get the most out of the pass as I believe I had to pay $13 for a return ticket to ferry to Manly.
Darling Harbour is awesome and the aquarium and wildlife centre are fun too. They have a little kids train that goes around Darling Harbour that looks cute and there are some water fountains there that the kids here are fascinated with.
The botanic gardens near the Sydney Opera house are lovely too.