In the myth of Theseus, the hero faces several challenges, including battling the Minotaur in the labyrinth, navigating deception from Ariadne, and dealing with the burden of his father's suicide after abandoning the black sails. These trials test his courage, intellect, and ultimately define his hero's journey as he strives to deliver Athens from tyranny. Theseus embodies the qualities of a hero, facing external monsters and internal conflicts alike.