Not exactly right. The Eldar, or the highest "race," of the Elves traveled, following the summons of the Valar, from the East of Middle Earth to its West, where most of them took ship (or floating island) to Aman, where the Valar dwelt.
Many millenia later, most of the Noldor, one of the Eldarin tribes, rebel against the Valar and return to Middle Eart to get revenge on Melkor/Morgoth. (Not a particularly good idea, as it turns out.)
Some of the Noldor also wanted to be rulers and not under the tutelage of the Valar anymore. One of these was Galadriel. Such desire was not necessarily evil in and of itself, althought the means used to achieve it certainly was.
Tolkien's conception of Galadriel changed throughout his life, so depending on which period of his writings you access, she is either denied permission to return to the West at the end of the First Age, or she refuses to return. In either case, her long rebellion against authority ends when she refuses to take The Ring.
BTW, not all the Elves went to Aman. Most of the Elves Legolas' family (and Galadriel) rules over are of this category. They are not nearly as high in moral and mental stature as the Eldar. Seemingly they don't differ tremdnsouly from Men, except in their longevity.
Similarly, the Numenoreans at their height were almost equal to the Eldar in power and status, except again for their lifespan. To me this indicates that Tolkien saw little difference in the raw potential of Men and Elves. The Elves mostly developed greater power because they had more time to do so in.
Even the Elves that had never been to Aman are fated to "fade" as the Third Age progresses. Tolkien implies that this is due mostly to the wearing effect the fire of their immortal spirit has on their bodies, which are made of the perishable stuff of Middle Earth. Only in The West can the Elves continue to live without this fading.
Although Elrond and Galadriel both imply that this "fading" is already underway, with a strong implication that it has been stopped or slowed during the Third Age by the power of the Three Rings (which the Elves can use as long as Sauron doesn't hold the One). With the destruction of the One Ring, the Three also lose their powers, so the fading will accelerate greatly.