Pre-school is the general term for schools which provide education for children before primary school.
CCC -> ChildCare Centre. Usually open from 7am to 7pm, some places offer additional
opening hours till 9pm for parents who work late. Basically is more like a nanny in school rather than an academic school.
Kindergarten -> A proper school which children only go to a few hours a day for proper instructional lessons. However, not all centres are academic focused due to MOE's instruction of 'learn through play'.
Playgroup is the age group. Usually from 18mths to 3yo in CCC.
There are also enrichment centres (these are not proper registered kindergartens) that offer playgroup programs. Some with parents involvement, some without. Different centres have different programs/focus. You need to check them out individually to understand what they are offering (can range from phonics to manadarin to music to sports).
Pre-nursery is also age group. Usually refers to 3yo.
Nursery is for 4yo, K1 for 5yo, K2 for 6yo.
In SG, formal education is compulsory AT PRIMARY LEVEL ONLY. Which means you can homeschool your kiddo till it's time to Primary school without any need for special permission or anything.
Before we can recommend anything, you need to identify what you need.
Do you need at CCC to look after your child the entire day or maybe half day?
Or are you just looking for somewhere that your child can attend a few hours of school a day to learn and interact?
Or are you looking for something academic?
It makes a huge difference cos I personally feel that my boy learnt alot of social skills at CCC but not much on academic. Now that he's 3yo, I changed him to a proper kindergarten that very academic focused.