Hihi my girl is also 2 1/2 months old, she'll be starting in infant care beginning from March as i'm returning to work. No one to help look after her (mum ill, mother in law passed away years ago) so no choice.
We actually registered her for infant care right after she was born and had her birth cert ready, cos there's usually a waiting list for most centres. You may want to start checking out the centres now? For certain places like Learning Vision, you can register even before your baby is born. Others may have different policies.
My baby is going to PCS, the branch at ghim moh. At first they only had an available slot in May, so hubby also enrolled baby at Learning Vision (in NUS - super ex, its slightly under 1k after subsidy). Luckily a slot came through for the PCS one which is cheaper (approx 600 after subsidy), so we cancelled the place in Learning Vision. They have not't refund us yet though.
I believe there's one infant care centre at Bukit Merah Central as well as another PCS branch at Holland. Not so sure about those in the west..