When you try to find the difference in words that seem similar - check an English/English Learner's Dictionary. They are better than a thesaurus in explaining the differences in words, because they are written especially for foreigners who need a guide to nuances in the language a native does not need.
'believe' - means 'to be quite sure or certain of something without having an actual proof of it.
In detail here are all possible angles to look at 'believe':
1 [transitive] - to feel certain
to feel certain that something is true or that somebody is telling you the truth
2 [intransitive, transitive] - think possible
to think that something is true or possible, although you are not completely certain
3 [transitive] believe (that)… - have opinion
to have the opinion that something is right or true
4 [transitive] - be surprised/annoyed
don't/can't ~ used to say that you are surprised or annoyed at something
5 [intransitive] - religion
to have a religious faith
'trust' - has a few meanings, the first given here compares most to 'believe'
The difference is that to trust someone you must have the 'belief' that this person will not harm you.
1 [uncountable] trust (in somebody/something)
the belief that somebody/something is good, sincere, honest, etc. and will not try to harm or trick you
2 [countable, uncountable] (law) an arrangement by which an organization or a group of people has legal control of money or property that has been given to somebody, usually until that person reaches a particular age; an amount of money or property that is controlled in this way
3 [countable] (law) an organization or a group of people that invests money that is given or lent to it and uses the profits to help a charity
4 [countable] (business) (American English) a group of companies that work together illegally to reduce competition, control prices, etc
"believe in someone/something"
can in its meaning be quite similar to 'trust'. It is a phrasal verb, meaning you can 'trust' someone to do the right thing and to be successful in what they do.
A second maning is incorporated here: to think that something is good, right or acceptable
The descriptions are a short form of the oup advanced learner's dictionary:
http://oald8.oxfordlearnersdictionaries.com/dictionary