I'm not sure I have a favourite bit of LotR: to me it feels somewhat like an arch, where the robustness of the whole structure depends on all of its parts. Extending that metaphor, the foundations must be the chapters "The Shadow of the Past" and "The Council of Elrond" - they explain everything that has happened before, and foreshadow and allude to everything that follows.
However, there are some bits which I seem to quote or think of particularly often:
Frodo and Gandalf in "The Shadow of the Past"
"I wish it need not have happened in my time," said Frodo.
"So do I," said Gandalf, "and so do all who live to see such times. But that is not for them to decide. All we have to decide is what to do with the time that is given to us."
And a few pages later:
(Frodo) "He deserves death."
(Gandalf) "Deserves death? I daresay he does. Many that live deserve death. And some that die deserve life. Can you give it to them? Then do not be too eager to deal out death in judgment. For even the very wise cannot see all ends."
Gandalf's outburst at Denethor in "Minas Tirith", ending with "For I also am a steward. Did you not know?"
