

All information comes directly from the writings of J.R.R. Tolkien. There are footnotes at the bottom of each section and a complete list of those writings used can be found here.
When Fëanor lost the Silmarils he led a revolt against the Valar, making powerful speaches to the Eldar. Many of the Eldar at that point didn't trust Fëanor, but that didn't stop them from wishing to see Middle-earth.
"Finrod was with Turgon, his friend; but Galadriel, the only woman of the Noldor to stand that day tall and valiant among the contending princes, was eager to be gone. No oaths she swore, but the words of Fëanor concerning Middle-earth had kindled in her heart, for she yearned to see the wide unguarded lands and to rule there a realm of her own will."1
When the Noldor went to the Teleri and asked for their ships so they could sail to Middle-earth, the Teleri refused and tried to persuade them to remain in Valinor. Fëanor then attempted to take the ships by force, but the Teleri resisted and a battle began. But Galadriel did not fight with the other Noldor, instead she fought for her mother's people.
Even though Galadriel took no oath and she didn't kill the Teleri, she still fell under the Doom of Manwë that forbade any of them to return to Valinor. The remaining Noldor who had not made it onto the ships of the Teleri were to proud to return and ask the forgiveness of the Valar, so they attempted to find another way across the sea.
"The fire of their hearts was young, and led by Fingolfin and his sons, and by Finrod and Galadriel, they dared to pass into the bitterest North; and finding no other way they endured at last the terror of the Helcaraxë and the cruel hills of ice." 2
"Even after the merciless assault upon the Teleri and the rape of their ships, though she fought fiercely against Fëanor in defense of her mother's kin, she did not turn back. Her pride was unwilling to return, a defeated suppliant for pardon; but now she burned with desire to follow Fëanor with her anger to whatever lands he might come, and to thwart him in all ways that she could." 3
1 J.R.R. Tolkien, The Silmarillion (New York: Ballantine Books, 2002) 90.
2 J.R.R. Tolkien, The Silmarillion (New York: Ballantine Books, 2002) 98-99.
3 J.R.R. Tolkien, Unfinished Tales of Númenor and Middle-earth (New York: Houghton Mifflin Company, 1980) 230.