"Trust" is a more general term. It means to rely on someone or something or to believe that someones actions in general are trustworthy. "Believe" is usually used for some statement or group of statements made by a person or group. This does not count the use of "believe" as judgement ("I believe it will rain today." Examples: "He is one of the few politicians I trust." "I trust her to take care of my children when I'm gone." "He says he made his best effort and I believe him." "I believe what the Bible has to say." You can also say "I believe in him/her/it", which usually indicates a deep trust.

pronunciations

Normally trust and believe mean two different things. However, in this case, the context I think you mean is like when someone says "Trust me, he's never going to get married" or "Believe me, I've tried..." In both cases, it emphasises the point. The meaning is similar I think to 我告诉你。