Well, right from the start we know there's a lesson to learn, and we know that someone is going to die. Heavy. But the title, A Lesson Before Dying, doesn't tell us who is going to learn the lesson.
At first it seems like we're talking about Jefferson, because everyone is obsessed with sending the teacher to teach Jefferson a lesson about Becoming A Man. And Jefferson does, in fact, come of age before his untimely death. But it also seems like the title allows for us to understand that Grant, too, learns a lesson.
Grant doesn't believe that it is possible to teach Jefferson to become a man before he's sent to the electric chair. But the ending shows Jefferson stating, as his last words, that he is a man, and has learned to stand on his own two feet no matter how terrible the circumstances are. This softens Grant's cynical nature.
Grant ends up learning a lesson about being a man too, but it has more to do with being willing to show weakness than acting strong. When Grant cries in front of his students after Jefferson's death he is finally letting them see that he is vulnerable, and that he has feelings.