You could still bring your big buggy as that would be nicer for the East Coast walks but just have a smaller buggy like a Maclaren for shopping and trips on public transport.
I've been using the buses, taxis and MRTs for the last 2 years, first while pregnant and now with a 17 month old. Most of the comments on here are true but it's tolerable and if your expectations of people start off low to begin with, you won't be disappointed to be expecting things to be different.
I live on the East Coast and if I'm going to somewhere like Marine Parade, I would get a taxi and sit child on my lap. If going to nearest MRT station, would pay $4 to get a cab to the station then MRT around town for the rest of the day. There is a shuttle bus from my condo
to the MRT station but as my child has only just started walking, this wasn't a good option for me.
I have done buses but that for me was the biggest nightmare because you can really only wheel your stroller on the newer buses which have disabled access at the back (where people get off). It's nice and wide and lowers to the kerb, however the driver often leaves a 6 foot gap between the bus and kerb so you end up having to carry the stroller on anyway. The older buses are just not worth it unless your baby is able to walk and you collapse the stroller but I've often had the driver wave his finger at me tell me to get the next bus.
We do have a car now but it's taken 2 years to be able to put the money together for the deposits etc. Public transport is not impossible but you need to have a relaxed frame of mind and not be stress head like me