I'm not sure if I should use "a" or "the" in this sentence. It's the first sentence of the story.
"While waiting for her boyfriend, [a/the] woman saw..."
If it's the very first sentence, then it's up to you. However, "the woman" would suggest to me that the rest of the story/book/novel is going to be based around that woman - it makes her sound important. "A woman" might mean that she is a peripheral character and you are just trying to set a scene in the reader's head. By the same token, "a woman" might later turn out to be very important. In this case, it's really your choice.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.