What does it mean to ride someone's coat tail?

It means to benefit from the success of another. A son may have it rather easy in life, for instance, because of his father's fame, prestige, or wealth. It can also be used in a contrasting sense, like a pastor's child not being able to ride on their father's coattail to Heaven, meaning that God judges everyone based on their own merits, not those of another.

A related idiom is "getting in on a Lewis." That means being accepted because someone in the family was accepted. That came from Freemasonry. They don't usually accept members under a certain age unless their father is in good standing.