Here are some examples to show the differences:
-- I trust you, so I will loan you $10. (Trusting in someone means I assume they will do the right thing. In this case, I assume you will pay it back my $10.)
-- I believe you when you say you’ll pay me back, so I will loan you $10. (Believing someone means assuming that they are telling the truth.) You might not trust someone, but you might believe something they say.

-- I believe in you, so I know you’ll succeed at school. (Believing in someone means you assume they have the skills or talent or ability to accomplish what they want to accomplish.)
-- I believe in ghosts, so I’m afraid of the dark. (Believing in something supernatural means you think that it is real.)
-- I believe in god, so I go to church. (Believing in something religious that cannot be proven means you have religious faith in this thing.)

In spoken English, “trust” is sometimes used to ask a question when you assume the answer is “yes” (you would not use “I believe” in these cases).
-- I trust you’re enjoying your visit to this country? (This means the same as “I’m assuming that you’re enjoying your visit?” but it is more delicate a way to ask.)
-- I trust you slept well? (This means “I assume that you slept well, and I’m hoping that it is true” but it is more delicate a way to ask.)

In spoken English, “believe” is sometimes used to ask a question when you think there is a possibility that it is not true, but you think it is probably true.
-- I believe we’ve met before. (I think we met before, but I’m not 100% sure)
-- I believe that my name is on your list. (My name should be on your list but there is a chance it is not.)
-- I believe we have a reservation for 7pm.

In business settings, you can use “I believe” to soften an argument or to be more polite (you would not use “I trust” in these cases):
-- I believe the decision was to cancel this project.
-- I believe we need to settle the outstanding bill.