OP, you're getting some great help and info here.
I read about the lottery system for primary(!) schools now.

(Only used to be for the top secondary schools).
Reason is, like everywhere else, the recession is here too, the housing market is "shut", so no-one can afford to move and more people want to stay in Amsterdam with kids rather than move out to the suburbs (boring).
(I don't have my nickname for nothing LOL).
Wow, things have changed since mine went to school....(he's 2nd year uni now)....
I can absolutely recommend having your child at peuterspeelzaal, it's only half a day and you get to meet the local kids and their mums.
Besides, he will pick up the language in no time, (which is what he needs by the time he gets to school proper) and you will be immersed in it too which will help you get a "leg-up".
Don't worry, most people speak English anyway, so you won't be floundering.
My son is still best friends with some of those he went to peuterspeelzaal with, and the circle just gets bigger once they go to primary and secondary.
He has a huge social circle now.
Main thing tho, is get your housing sorted. Everything follows on from there. Once you are registered in your house you can register in the area for everything. (Includes GP, etc.).