If you believe something, you believe it is true. But there are many things that I believe are true which are actually false.
I cannot believe something is true and also believe it is false. (Though I have some doubts about that statement, since while it sounds quite reasonable, I often act as if I believe something is true when in further reflection, I would assert that I was in error.)
But, if I belief something, I believe it is true. And if I believe it is true, then how can I come to know that it is, in reality, false?
I could learn from arguments, from experience, and experimentation. I could learn from evidence from the outside world, or from reasonable deduction in my own mind. I do not need proof to change my beliefs; I just need some amount of persuasion.
But sometimes truth and error are hard to discover, because the whole way I perceive the world is not necessarily accurate. My own thoughts and mind serve as a filter.
There was a car that Dillon owned that I believed was blue and he believed to be brown; we went and found a picture of it to resolve our dispute. But on seeing the picture, Dillon still believed that it was brown and I still believed that it was blue.
So sometimes our perception of truth and error and beliefs are so complicated that it is in fact extremely difficult to say what is true and what is false, and to change our beliefs to be more accurate to what actually is.
But I do think there are things that are true and there are things that are false, and even if my beliefs will never be incredibly accurate, I do want them to be more accurate than they were yesterday.