Santiago proves to himself that his strength is intact. By catching conquering the big fish, Santiago shows that he is not too old to achieve significant victories. However, in losing the fish to the sharks, Santiago is reminded that the ocean (and Nature) is a much greater force than any man can reckon with.
Thus Santiago's pride in his determination and strength of will are justified, but he does not become a hero. He proves his capabilities but still suffers a defeat.
Additionally, Santiago's friendship with the boy is solidified. Though Santiago and the boy both respect the boy's father's wishes initially, the boy finally decides that he will make his own decisions. Friendship is more important than reputation and/or superstition.
These lessons are all inter-related, as Santiago also learns that not all defeat is absolute. There is some victory in his adventure, mingled with his defeat.
After his defeat he says the boy should not fish with him because “I am not lucky anymore.” Yet Santiago quickly changes his mind about going out with Manolin when the boy says that “we will fish together now, for I still have much to learn.”