What do Frodo and Sam teach and learn in Lothlorien? Who gets taught, and who teaches them?

In both the book and the movie, Frodo and Sam reach Lothlorien after passing through the Mines of Moria, where they are almost killed by orcs and attacked by a Balrog. They have lost Gandalf and are distraught, as well as exhausted.
They are met by Galadriel, who leads them to Lothlorien and offers them food and shelter. In the book, she meets with Sam and Frodo and allows both of them to look into her Mirror, which shows "things that were, and things that are, and things that yet may be."
Sam sees Frodo asleep under a cliff, whereas Frodo sees what seems to be Gandalf but dressed in white, the sea, and the Eye of Sauron. It is not obvious yet, but Sam and Frodo are both learning about their futures.
In the movie, only Frodo is allowed to look into the Mirror. Frodo learns how important his mission actually is. He has a vision of death and destruction; the Shire is destroyed and orcs rule Middle-earth. He learns that destruction awaits Middle-earth should his mission fail.