In "Hamlet" does Hamlet undergo any changes throughout the story? Does he grow psychologically? Morally? Spiritually?

At the beginning of the play, Hamlet appears to be a rather petulant and highly depressed individual.  Whether or not this is representative of is normal self is unclear--after all, his father has just died, and his mother has formed an "o'er hasty" attachment to his uncle.  These could give any normal person cause to whine and mope a bit.  So, taking that as a starting point, we can see some pretty significant changes in Hamlet throughout the play.

Psychologically, at the beginning of the play, he is tormented about how and if to enact revenge.  He is torn--he doesn't know what to do.  His conscience steps in, telling him that murder is wrong, then his emotions take over and tell him to just do it, and he battles himself for most of the play.  Psychologically, he ponders death and suicide, and his unenviable position.  But by the end of the play, he is at ease in his mind.  He has resolved that "the readiness is all," and is ready to enact revenge.  He is no longer conflicted, torn within himself, and going back and forth.

Morally, Hamlet also resolves his potential dilemma with murder.  He plays a role in the death of his good friends, and is ready to murder his uncle.  So, that could constitute a degradation of his morals, although he feels that it is being true to himself and what he knows is true about others.  Morally speaking, we see him turn from a bitter person to a ranting and accusatory friend and son; he takes the high ground against his mother and Ophelia, lashing out against them, and expressing great moral angst over the transgressions of women.  By the end of the play, he seems to be at peace with even this.  He no longer lashes out at his mother, and seems geniunely distressed over Ophelia's death.

Spiritually, Hamlet is unhealthily attached to the idea of death and Hell at the beginning of the play. He can't commit suicide for fear of repercussions in the next life, his father's ghost might be a "demon from Hell" come to tempt him to murder, and he can't kill his uncle while his uncle is praying because that might send him to heaven.  By the end of the play however, he has put aside his spiritual qualms and decided to act, even if it means dying.  He resolves to be strong, and not let spiritual excuses stand in the way of action.

I hope that those thoughts helped; good luck!