how does Nietzsche challenge our value system

Much of our value system in the US is based on Enlightenment and Christian thinking, both of which Nietzsche sharply challenges.
Nietzsche coined the term "God is dead," a direct challenge to Christian thought. By this, he meant that the concept of God that for a millennia held together the institutions of Western society was no longer viable in the modern age. We just don't believe the way we once did, he said, and so have to devise new ways to cement society together.
Nietzsche also challenged the Christian theological concept that the "meek shall inherit the earth," and opposed seeing mercy, forgiveness, and compassion as the highest moral values. He called Christianity a slave mentality based on the weak—of whom there are many—resenting the strong, of whom there are few. He envisioned a society that would value the strong, the "übermensch."
Nietzsche challenged the Enlightenment idea that ethics and morality are universal and unchanging, arguing instead for a genealogy of moral/ethical ideas that unfolds based on who has power in society. He proposed the modern world going back to Greek morality, which, unlike Christian morality, was based on exalting strength and "master" or "übermensch" people, such as Achilles.
Nietzsche also opposed the Enlightenment idea that language is like a clear windowpane of glass illuminating truth. Under this theory, if people can find the right words, they can reveal truth in a transparent way. No, Nietzsche said. He described language instead as a "prison house." We are born into a language that limits how we think before we are even old enough to know what is happening. Rather than illuminate truth, the prison walls of language keep out truth.
As we can see, Nietzsche was a philosopher willing to upend the "givens" of Christian and Enlightenment thought.