No you cannot choose any school even if it's very near to home. When I changed my son's school from Marsiling to CCK, he got the one that is 2 stops away from our house instead of the one just next to our new block. Depending on the slots available. Not the ones near to home.
You have to inform the current school first that you want to change your child's school. They will pass you a form to fill in then have to give them back. Wait for the reply in your letterbox. The new school will contact you. Or you can also visit STEPS page on the MOE website then just follow the instructions there but to me informing the school first and get the form is faster.