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.” 

What lessons about life does Manolin learn from Santiago in The Old Man and the Sea?

Manolin (the boy) follows Santiago (the old man) because he sees that he is a man of honor and courage. Manolin knows that other fishermen have newer equipment, larger boats, and more men, but not the pure skill and learned experience that Santiago possesses. There is a difference between fishing for food and sale, and fishing because it entails the entire heart and soul of a man's life; Santiago's life is in his work:

"Que Va," the boy said. "There are many good fishermen and some great ones. But there is only you."

"Thank you. You make me happy. I hope no fish will come along so great that he will prove us wrong."

"There is no such fish if you are still strong as you say."

"I may not be as strong as I think," the old man said. "But I know many tricks and I have resolution."
(Hemingway, The Old Man and the Sea, Google Books)

By the end, Manolin sees how Santiago has refused to give in, refused to let the sea and the ignorance of others influence his actions. Manolin learns the value of persistence, of pride (in its proper place), and of willpower over weakness. Manolin resolves to learn everything that Santiago knows so he can become as great a man as Santiago, and so he can pass those lessons on to his own children in the future.

What can we learn from Stantiago in The Old Man and the Sea?

I think one of the definite lessons that we can learn from the character of Santiago is the way that he is determined to continue fishing, in spite of his crushing and massive disappointment at having lost the big fish that he caught. Although he is shown to have given up so much and sacrificed almost everything to capture this fish, and in spite of his every efforts, he cannot save the fish from being eaten by the sharks that surround him. Yet Santiago is not overwhelmed by this failure. When he gets back, after some encouagement from Manolin, he begins to talk again about going out the next day. The very last line of the novel is very significant, as it indicates how Santiago, in spite of his failure, has not abandoned his dreams:

The old man was dreaming about the lions.

The lions refers to a dream that Santiago has had to go to Africa and see the lions. The fact that the novel ends on this sentence is very significant as it indicates that Santiago has transcended his failure and is not crushed by it. If anything, he is even more determined to go out and fish again. This is a lesson we can all learn from.

Yet the struggle to achieve one’s dreams is still worthwhile, for without dreams, a human remains a mere physical presence in the universe, with no creative or spiritual dimension. And so at the end of the story, Santiago, in spite of his great loss, physical pain, and exhaustion, is still “dreaming about the lions”—the same ones he saw in Africa when he was younger and would like to see again.