Williamsburg schools ~ toddler to kindergarten

I’m considering moving to Williamsburg and also planning for a baby. Trying to research/understand the new york school system is confusing to say the least. My understanding - please correct if this is wrong - is that we will have to pay for private childcare until kindergarten at which point they’ll be eligible for public school. Before that, childcare options are only for low income or on a low chance lottery.

Based on that, I’m trying to figure out what are the good daycares in Williamsburg. Are there daycare rankings? (not sure if that’s an absurd question)

I was looking at Williamsburg Montessori and Ms. Lam Montessori. Does anyone have experience with either of those schools?

Any insight on public elementary schools would also be appreciated!