what are the moral lessons of the story?

There are certainly quite a few "moral lessons" to be gleaned from the story of Dmitri Karamazov. To answer specifically, however, I would draw attention to the radical vulnerability that his relationship with Grushenka eventually exemplifies at the end of the story. The characters' warring egos (Dmitri vs. Fyodor Pavlovich, Grushenka vs. both men) rage on until they find themselves in the profound human situation of having nowhere to turn. As Alyosha's patient life witnesses, there is no room for bourgeois guardedness in any Dostoevskian account of authentic love. There is only mutual vulnerability and suffering.