Hi Myermay,
I live in Sydney & have 3 children (youngest is 10mths). I rely on cabs as a regular mode of transport as well as buses & trains. I'm lucky to have 4 regular drivers who I can call. Prebooking thru the companies has been extremely frustrating as they don't guarantee they can get you a cab & I have been stranded at the last minute (cabbies are bad for being lazy). However I have found Premier cabs to be good.
There is a train at the airport, fast to city but expensive($12 o/w). At the airport you can ask the taxi council assistant to get you a cab with child seat (both capsule & seats for older children are available. You dont need to pre-book as cabs are plentiful. Look out for the wheelchair cabs - the drivers are excellent & the seat is usually already set-up. Sometimes you pay for the driver to put the seat in (meter turned on).
Also be aware the advice given about having your son on your lap (Perth) IT IS ILLEGAL. Children 4 years & under must be in an approved child restraint in all cars, vans & taxis.
Public transport is good, some buses are wheelchair accessible and you can wheel a pram or stroller onto the bus. Buses that are wheelchair friendly have the letter "a" on the timetables next to the times of the bus, although this is not always reliable. The bus also has a small wheelchair symbol on front window near doorway.
Trains are the same, some stations have lifts/ramps or escalators. Staff are not allowed to assist you carring prams/strollers up & down stairs.
I highly recommend bringing an umbrella stroller with you, I use the Maclaren Vogue. A damned good investment, especially with a hood because it is very hot in summer. You can buy a very cheap maclaren just for travel - safe, strong, reliable & so easy to fold.
Good luck with your travels & I hope you enjoy Australia.