Learn German: Du kannst mir trauen.

The verb trauen (to trust) is a strict dative verb. In English, we say "trust me" using a direct object, but German requires the dative case here. Therefore, you must use the dative pronoun mir (to me) rather than the accusative mich (me).

Why is the verb trauen at the very end of the sentence?

This happens because of the modal verb können (can). In German, when you use a modal verb, it gets conjugated to match the subject (Du kannst) and sits in the second position. The main action verb (trauen) gets pushed to the very end of the sentence in its basic dictionary form (the infinitive).

How would I say "You can trust him" or "You can trust us"?

You just swap mir for the correct dative pronoun. "You can trust him" is Du kannst ihm trauen. "You can trust us" is Du kannst uns trauen.

Could I use glauben instead of trauen here?

You could, but the meaning changes slightly. Du kannst mir trauen means "You can trust me" (as a reliable person, or with a secret). If you say Du kannst mir glauben, it means "You can believe me" (that what I am saying right now is true). Conveniently, glauben is also a verb that always takes the dative case!