It was in the days after Galadriel was born that Melkor - still appearing as a Vala unparalleled in might and splendor save by Manwë and Varda - walked the streets of Tirion upon Tuna, poisoning the hearts of the Noldor against each other and against the Valar. But it was by no device of his that Fëanor the Mighty then succeeded in capturing the light of the Two Trees in the three Silmarils. Shortly thereafter Melkor with help of Ungoliant slew the trees and Valinor was darkened. And when the Valar asked Fëanor for the light of the Silmarillii to rekindle the light of the trees, he refused.
Melkor slew Finwë, high-king of the Noldor and took all the jewels from his house in Formenos, and the Silmarils were gone. Melkor escaped from Valinor and set up his refuge under the peaks of Thangorodrim, in the ultimate north of Middle-earth. And in his crown he set each of the three Silmarils.
In Tirion the Noldor came assembled when Fëanor called Melkor by the name of Morgoth, vowing to pursue that foe until the end of times and beyond. The Noldor of his house then took the Oath, unbreakable and irrevocable, by which they were bound to pursue with hatred and vengeance any Vala, Demon, Elf or Man who possessed a Silmaril. Though Fingolfin, Turgon, Finarfin and many others spoke against such an oath, some kindly, others fiercely, each and every Noldo was then set to leave Tirion and Aman and go in pursuit of Morgoth the Accursed.
Galadriel swore no oath, but eager to go she was nonetheless. And even when the herald of the Valar bade them not to pursue, they left Valinor, the great host of all the Noldor in their splendor, some hurriedly in front, with Fëanor and his sons, some thereafter in the company of Fingolfin and his son Fingon, and others at the back - looking back at times at the lamps of Tirion - with Finarfin and Finrod.
Thus it was that the host of Fëanor reached Alqualondë before the others. Fëanor then claimed the help and ships of the Teleri for the purpose of sailing east. But Olwë and his people refused, for the ships were precious to their hearts, but also they trusted the judgment of the Valar. Then the Noldor slew the Sindar of the Blessed Isle, taking their ships by violence, and alas, in that battle - be it by confusion or intent - they were aided by the vanguard of the house of Fingolfin.
The books say not in which host of the Noldor Galadriel partook, but it is probable that she went with her brother and father and thus came too late to prevent the Kinslaying. But it is also possible she went with the sons of Fingolfin, her friends and of like mind in the matters that had passed, in which case she undoubtedly would have done all she could to defend the people of Olwë, her grandfather and of Earwen, her mother.
The Kinslaying of Alqualondë caused the Wrath of the Valar. Mandos came to them and spoke the Prophecy of the North, the Doom of the Noldor, cursing Fëanor and all who followed his path, foretelling that their actions would have no results but misery and grief. It was then that Finarfin repented, and returned to Valinor with many of his host, and they were pardoned and allowed back into Aman. But the children of Finarfin did not follow them.
Why did Galadriel and her siblings not go with their parents? Their friendship with the sons of Fingolfin was undoubtedly a reason. Surely in their youth and confidence they were proud and self-willed too. But an important reason also, no doubt, was their sense of responsibility. Finrod, Orodreth and especially Galadriel did neither trust nor like Fëanor, and they would not leave the fate of their kinsfolk into his hands alone. But by not returning to Valinor their path was set: to fall under the doom of the Noldor and its prophecy of tears.
After the Prophecy of Mandos Fëanor and his people sneakily depart from the Isle, leaving the hosts of Fingolfin and Finrod behind. And when they hit landfall in North-west Beleriand, they burn the ships instead of sending them back.
Thus the remaining Noldor, led by Fingolfin and his sons and by Finrod and Galadriel, were forced to take the northern route by ice from Araman to Beleriand: the Helcaraxë. A great force they were, but the terrors of the Grinding Ice were perilous, and many were lost. But at last they marched into Middle-earth and there they were greeted by the first rising of the moon.
 In the Silmarillion it is said that also Feänor’s words had “kindled” in Galadriel’s heart a longing for strange lands to seek dominion of. This Tolkien in hindsight might want to have altered or even deleted this if he had had the opportunity. For more and more Galadriel develops – in his later writings – to Feänor’s ultimate opposite and antagonist.