The fellowship leaves Lorien with personal knowledge gained and group consultation given. In Gandalf's absence, leadership falls to Aragorn, even if his leadership is neither to Boromir's or his own liking. I get the strong impression that Aragorn does not wish to bear the mantle of leadership, adding into his gratitude for the boats and a direction of travel that does not require a decision yet on his part.
The role of the Ring and that of Frodo as the Ring-bearer have weight that's beginning to show. Clearly Boromir finds the temptation to use the ring against Sauron a compelling one, saying "folly to throw away," before catching himself. The direction of the Fellowship lies in if they seek alliance or to steal directly into Mordor, and at every encounter the threat by the mere presence of the Ring grows.
The elves are quite generous in sending the Fellowship off. Lembas, fair cloaks, boats, and even rope. They also provide the gift of song, with Galadriel singing them down the river and a final meal. I find this a touching and somewhat tender gesture, one more pronounced for the already apparent diminishing of Galadriel, seen by Frodo as "present and yet remote, a living vision of that which as already been left far behind by the flow of Time."
In this last parting meal, Galadriel and Celeborn also bear blessed gifts, with a scabbard reminisent of Excalibur's and a token of hope from Arwen Evenstar for Aragorn, precious belts to Boromir, Merry, and Pippin, a bow to Legolas, earth from her garden for Sam, three strands of her hair for Gimli, and the bottled light of the Earendil star for Frodo. Some gifts may seem odd to the observer, but are perfectly matched to the receiver, whether they know it yet or not. Gimli's request and Galadriel's acquiescence are more than a courtly gesture, but a token of faith, friendship, and respect between two that should be estranged. Frodo's gift will serve him well in the face of the encroaching darkness, but it's Sam's gift that is perhaps both seemingly the most mundane and the most touching.
I wonder at the level of deliberation in boat organization. There is wisdom in putting Frodo (and Sam) with Aragorn, but I wonder how aware members of the party are of Boromir's growing disquiet. Either Gimli or Legolas would also be well matched with Frodo, but their developed friendship is quite dear. I also wonder at elves speaking of stranger rumors of forest dangers that they do not fully comprehend. Armed with foreknowledge I know they speak of the ents, beings long of Middle-Earth, yet in this time seemingly alien to the elves. It seems that the elves would have a closer relationship with the ents, but perhaps time and strangeness has caused distance between to grow.
The film inserts some extra footage here, in particular a very visually striking scene with Saruman and an uruk-hai. This was kept relatively brief, but shows us the danger literally chasing the Fellowship. The actual farewell and gifting has been largely excised, but we see Galadriel hand the vial to Frodo and watch them leave. As they solemnly paddle down the river, footage of running uruk-hai is cut in while an increasingly martial soundtrack swells.