Here’s the plot summary from Wiki:
Comprising two distinct but interrelated plots, the narrative runs back and forth between the two, taking up each plotline in alternating chapters.
The odd chapters tell the 15 year old Kafka’s story as he runs away from his father’s house to escape an Oedipal curse and to embark upon a quest to find his mother and sister. After a series of adventures, he finds shelter in a quiet, private library in Takamatsu, run by the distant and aloof Miss Saeki and the intelligent and more welcoming Oshima. There he spends his days reading the unabridged Richard Francis Burton translation of A Thousand and One Nights and the collected works of Natsume Sōseki until the police begin inquiring after him in connection with a brutal murder.
The even chapters tell Nakata’s story. Due to his uncanny abilities, he has found part-time work in his old age as a finder of lost cats (a clear reference to The Wind-Up Bird Chronicle). The case of one particular lost cat puts him on a path that ultimately takes him far away from his home, ending up on the road for the first time in his life. He befriends a truck-driver named Hoshino. Hoshino takes him on as a passenger in his truck and soon becomes very attached to the old man.
Nakata and Kafka are on a collision course throughout the novel, but their convergence takes place as much on a metaphysical plane as it does in reality and, in fact, that can be said of the novel itself. Due to the Oedipal theme running through much of the novel, Kafka on the Shore has been called a modern Greek tragedy.