He won't go today to school because he has an appointment.
or
He won't go to school today because he has an appointment.
adjuncts
He won't go today to school because he has an appointment.
or
He won't go to school today because he has an appointment.
Best Answer
These are the ways you could place "today" in this sentence:
and awkwardly (but grammatically):
As for the difference between doesn't and won't go to school, that should be asked as a separate question.