How does Siddhartha learn to experience the troubles caused by the life cycle?
The Four Noble Truths of Life are: 1. Existence is suffering 2. Suffering arises from desire 3. Suffering ends when desire ends 4. The way to end desire is to follow the 8-Fold Path to Nirvana (a. right belief b. right resolve c. right speech d. right conduct e. right occupation f. right effort g. right contemplation h. right ecstacy).
I think you are asking about the second and third Noble Truths. Siddhartha suffers because he desires knowledge. It is during this time in his life that he pursues knowledge outside of his life as a Samana, and he follows Gotama. He realizes that the Samanas have taught him many things, but have not led him to Nirvana. After following Gotama for awhile, Siddhartha realizes that Gotama, the Illustrious One, still does not have what Siddhartha needs. He says, "I must judge for myself. I must choose and reject." Through his conversation with Gotama, Siddhartha realizes that Gotama's teachings also will not lead him to Nirvana. He stops seeking knowledge from someone else and learns to depend on his own judgment which means Siddhartha will need to move on in his life, away from Gotama and the other monks. He will leave Govinda, his shadow and friend, there with Gotama. So, in doing so, Gotama has robbed Siddhartha of his friend, but has given him the gift of the knowledge of self. Thus, Gotama has helped Siddhartha wipe out desire, and with it, suffering.