a person is always lying about where he is going to be or what he is going to be doing. So, when he tells me something, I don't believe him anymore. What is a word that describes me?
You are distrustful.
distrustful: unable or unwilling to trust; doubtful; suspicious (Random House)
Distrust vs. mistrust
Distrust and mistrust are roughly the same. Both refer to (1) lack of trust, and (2) to regard without trust. But distrust is often based on experience or reliable information, while mistrust is often a general sense of unease toward someone or something. For example, you might distrust the advice of someone who has given you bad tips in the past, and you might mistrust advice from a stranger. This distinction is only a general tendency, though, and it is not always borne out in real-world usage.
There are a number of possibilities.
Dismissive describes your reaction to his statements.
Incredulous "unwilling or unable to believe something" is a good start, but the lack of belief could be caused by any number of reasons.
Profoundly mistrusting would describe your mental state, although it is two words rather than one.
Disillusioned does not precisely state that you do not believe the person, but it is pretty well implied.