Writing in Egypt goes back to pretty much the earliest writing anywhere. Nobody really knows yet whether the Egyptians figured out how to write for themselves, or whether they learned it from the Sumerians, who also began writing about the same time, about 3000 BC. The Egyptian form of
writing, hieroglyphs, does not look the same or work the same as the Sumerian form of writing, cuneiform. So if they did get the idea from the Sumerians, the Egyptians certainly changed it a lot.
What we have left of Egyptian writing, like Egyptian art, mostly comes out of tombs. Because of this, most of what we have left is prayers (because that is the kind of thing you put in people's tombs). Other kinds of writing like laws, letters to your mom, and lists of who gave their fair share to the temple mostly have rotted away, over the years.
The most famous prayer from Egyptian tombs is the Book of the Dead. The Book of the Dead was a set of instructions for how to get from the world of the living to the world of the dead. In the New Kingdom, starting about 1500 BC, people put copies of the Book of the Dead in their tombs to help them along the way.
One well-known story whose earliest version comes from ancient Egypt is the story of Cinderella.
To read actual stories from Ancient Egypt, try this book:
Tales of Ancient Egypt, by Roger Lancelyn Green