"Whether" is nearly always replaceable by "if", for embedded questions. So
I don't know whether Peter will come to school.
and
I don't know if Peter will come to school.
have exactly the same meaning: I don't know if it is true or false that Peter will come to school.
Your other sentence,
I don't know that Peter will come to school.
is grammatical, but in principle has a different meaning (though that meaning is unlikely with a first person subject).
If we switch to a third person subject:
Jane doesn't know if/whether Peter will come to school.
means that she doesn't know the truth of the question. But
Jane doesn't know that Peter will come to school.
implies that it is a fact (known to the speaker) that Peter will come to school, but that Jane doesn't know that fact.
Clearly, this distinction is unlikely for a first person subject, because the subject is the speaker. For this reason, I think, you do hear
I don't know that Peter will come to school.
with the same meaning as if/whether; but I would say this is very colloquial.