I think Jerry learned more than only two lessons in the story.
One lesson that he learned was the importance of training. I think he always knew the importance of training, which is why he worked so hard to build up his lung capacity and ability to hold his breath for long periods of time.
That day and the next, Jerry exercised his lungs as if everything, the whole of his life, all that he could become, depended upon it.
His training paid off and Jerry was able to hold his breath for the necessary two to three minutes
“Mummy,” he said, “I can stay under water for two minutes, three minutes, at least… It came bursting out of him.
Related to the lesson learned about training is what the training is symbolic of. The second lesson that Jerry learned is the importance and power of his own determination. He goes about his goal with a single minded focus. He achieves success, and Jerry no longer feels a deep need to prove himself to the other boys.
It was no longer of the least importance to go to the bay.
Why would it no longer be important to be with those other boys? After all, he screamed at the top of his lungs for them to simply notice him earlier.
And now, in a panic of failure, he yelled up, in English, “Look at me! Look!” and he began splashing and kicking in the water like a foolish dog.
It's no longer important because the lesson that Jerry learned is that what others think is unimportant. He met and achieved his own goal. He proved it to himself and that is much more important to him than anything else.
I believe that Jerry learned one more lesson. I believe that Jerry learned that independence from his mother doesn't mean that she must be absent from his life. From the beginning of the story, Jerry desperately wants to assert his independence from his mother. He does this by spending as little time with her as possible. When she forces Jerry to stay put, it chafes and grates on him. He doesn't take it well.
Again his nose bled at night, and his mother insisted on his coming with her the next day. It was a torment to him to waste a day of his careful training, but he stayed with her on that other beach, which now seemed a place for small children, a place where his mother might lie safe in the sun. It was not his beach.
But by the end of the story, Jerry has stepped across a threshold to manhood, and he no longer sees his mother as an enemy combatant.
I don’t think you ought to swim any more today.” She was ready for a battle of wills, but he gave in at once.
Previously, Jerry would have fought his mom on that comment. But since learning those other things about himself, Jerry doesn't feel the need to establish his independence by distancing himself from his mother.