In NY, there's no public (public = state) pre-school provision to speak of until pre-K which is the equivalent age to reception but in the US is not proper school age. Some public schools have pre-K but it is very oversubscribed and you shouldn't have high expectations of getting a place. Especially if you are moving from abroad since you'll have to live in the right district in March to apply for a September place. So you should expect to pay for pre-school if you want it probably.
From Kindergarten (age 5) you can go to your local public school. Mostly, you are zoned for a specific school depending on where you live so there's not the same choice that you would get in the UK. However this means that you can pick where to live and pretty much be guaranteed a place at your chosen school - although there are problems with overcrowding so it's possible for that to not work out in practice.