Providence schools has posted a schedule on-line and instructions for registering your child for kindergarten. These can be found at:
Note that if you are seriously interested in your child attending your neighborhood Providence Public school, you stand the most chance of getting your child into the school if you apply when your child enters Kindergarten. Once seats are filled, regardless of whether a child is from the neighborhood or outside the neighborhood, the only way to get your child into your neighborhood school (eg. in first grade) will be to fill seats empty by attrition.