Learn Russian: Никто не знает моё новое расписание.

Yes, никто means "nobody". But Russian requires double negation. Whenever you use a negative word like никто, ничего (nothing), or никогда (never), you must also put the negative particle не before the verb. Literally, the Russian sentence says "Nobody does not know my new schedule."

What verb form is used after никто?

You always treat никто as if it were the pronoun он (he). That is why the verb знать (to know) takes the third-person singular ending, becoming знает.

Why do we use моё новое instead of мой новый or моя новая?

The word for schedule, расписание, ends in -е, which makes it a neuter noun. Because of this, the words for "my" and "new" must also be in their neuter forms to match the noun's gender: моё and новое.

What case is моё новое расписание in here?

It is in the accusative case because it is the direct object (the thing being known). For inanimate neuter nouns like расписание and their matching adjectives, the accusative case looks exactly the same as the basic dictionary form.