We are still on the conservative side on breastfeeding. you can breast feed your kid in a corner properly covered up n others will not notice, otherwise there are nursing rooms in most of the shopping malls here.
Our public transport system is fairly well, waiting time should not be too long during peak hours n may be a little longer when its off peak. If you r unsure which bus stop to alight, you can approach our friendly bus captains. If you r lost in the MRT stations, ask the station control officers or any commuters, we are willing to lend you a helping hand.
As for car, a car seat is a must for kids. I dont drive so I am unable to give you info on hiring a car seat.