when we went to Melbourne we booked a limo with a car seat, it didn't cost us all that much more for having the piece of mind of 1- a driver who knew WHERE he was going( a rare thing in Melbourne so we discovered) and 2- the security of having a car seat (I am a particuarly paranoid mum when it comes to this)
hope you have a great holiday
I forgot to add that this was just to and from the airport the rest of the time we either walked or took a tram so I am not much help here sorry