I am far from certain, but I think you might be remembering "By the Waters of Babylon" by Stephen Vincent Benét. (Click on the title and it will take you to an online copy of the story which I found just now.) One reason I nominate this one is that Benét's stories are more likely to be collected in literature textbooks than is the average science fiction story about a post-apocalyptic world.
To provide an example of his lasting impact on other people's storytelling efforts: One of Benét's other classics, the plot of which has been adapted and imitated many, many times, in movies, TV episodes, etc., was "The Devil and Daniel Webster." (If you don't know what that story's plot was all about, then click on the title to read the story's Wikipedia entry, and you'll get the idea. I suspect that it will strongly remind you of other things you've run across at one time or another in your life -- TV episodes, for instance, as I mentioned.)
On a similar note, "By the Waters of Babylon" was published before it became a standard trope of science fiction to have a young protagonist roaming around in a primitive world which used to be the home of a high-tech civilization.
Here are some of the points which made "By the Waters of Babylon" pop into my head as I read your description from memory.
- It quickly becomes clear that John (the narrator) lives in what we would call a primitive culture which lacks the benefits of modern science. For instance, while describing how his father (the "priest" of their community) trained him to someday be a worthy successor in that important role, John says:
I was taught the chants and the spells -- I was taught how to stop the
running of blood from a wound and many secrets. A priest must know
many secrets -- that was what my father said.
If the hunters think we do all things by chants and spells, they may
believe so -- it does not hurt them. I was taught how to read in the
old books and how to make the old writings -- that was hard and took a
long time. My knowledge made me happy -- it was like a fire in my
heart. Most of all, I liked to hear of the Old Days and the stories of
the gods. I asked myself many questions that I could not answer, but
it was good to ask them. At night, I would lie awake and listen to the
wind -- it seemed to me that it was the voice of the gods as they flew
through the air.
We are not ignorant like the Forest People -- our women spin wool on
the wheel, our priests wear a white robe. We do not eat grubs from the
trees, we have not forgotten the old writings, although they are hard
to understand. Nevertheless, my knowledge and my lack of knowledge
burned in me -- I wished to know more. When I was a man at last, I
came to my father and said, "It is time for me to go on my journey.
Give me your leave."
That last bit is setting things up for the rest of the plot. The tradition is that a son of a priest, who wishes to become a priest, must go off alone on a special journey. However, you are not supposed to cross the river to the east, and it is forbidden to visit the area called "The Place of the Gods." John, however, feels that signs and omens are steering him in that direction, so he finds a way across the river. He calls it "Ou-dis-sun," which appears to be a degenerated version of "Hudson."
John builds a raft and uses it to go downriver until he can reach the other side of the river. He ends up in an area that is an abandoned city, full of huge buildings (skyscrapers, though he doesn't know to call them that), which is strongly hinted to be Manhattan Island, part of the former New York City. (I just checked: In the final paragraph of the story, we are finally told flat-out that another name for this ancient place is "newyork." At that point, the author is no longer hinting.) The sheer scale of those buildings is what makes John's tribe (and any other surviving tribes in the region) believe that the Gods themselves must have lived here, and built those colossal structures, ages ago, before some catastrophic event which is now remembered as "the Great Burning."
After finding the mummified body of one man in the city, John has an epiphany as he suddenly realizes the builders of this great city were not Gods -- they were just men, much like himself, except that they knew how to build great things, and then something went terribly wrong and they essentially destroyed themselves. (Benét never goes into detail about how this mysterious "Great Burning" happened, exactly, since John doesn't know much more about it than the legendary name. And this story was first published in 1937, long before anyone had ever built a nuclear weapon, but the story is now generally interpreted as being a "post-nuclear war" tale.)
So, except for the protagonist being a young man instead of a young woman (and you indicated you were not sure of the gender), I think this is a decent fit for what you remember once reading in a schoolbook. And, as I said, it is likelier than most post-apocalyptic stories to have been included in such a book.