In most cases the word ‘that’ is optional; you can put it in or leave it out without affecting the meaning. The shorter the sentence, the better it is to leave it out. So I would say:
She said she was going to the store.
In longer sentences, it is usually clearer to put it in. For example:
She told all her friends at the office that she was going to the store.
Sometimes you have to put it in to avoid ambiguity, e.g:
She said that at 3 o’clock she was going to the store.
or
She said at 3 o’clock that she was going to the store.