The white pioneers who moved from the American East Coast came from two different major groups. I will address the first group, about which I know comparatively less (and because you didn't really ask about them), before moving to the second.

The first group was of migrants from Europe. A great number of migrants left Europe after 1848 under the twin influence of potato blight, general population pressure, and the disruptions caused by the (mostly failed) revolutions that swept the continent in that year. In Germany, France, Hungary, and various other places groups that favored more liberal, republican forms of government attempted to overthrow monarchical rule and were, by and large, crushed. This resulted in a great deal of repression by the surviving regimes. I am the most familiar with mid-century Prussia, where a government favoring German unification under a liberalized regime briefly convened in Frankfurt before being summarily crushed by Friedrich Wilhelm IV, who then instituted a sweeping series of conservative reforms that included censorship and a bolstered secret police. But this happened around Europe, and migrants left for other reasons as well. Many of these people went West, populating the Great Plains. That's why there's such a wealth of Northern European culture in states like Minnesota, Wisconsin, and the Dakotas, where many Germans and Scandinavians ended up (though others moved further West and South, as well).

But, you were asking specifically about a native-born East Coaster who moved West. The primary drivers of Western migration by American citizens were both a "push" and a "pull." First, the "pull," because it's a bit shorter. The United States Government had obtained a ton of land in the Treaty of Guadelupe Hidalgo that ended the War with Mexico, as well as the subsequent Gadsden Purchase and the treaty with England that set the Oregon border, to go along with land it had already acquired in the Louisiana Purchase and after independence. By the time of the Gadsden Purchase, in 1853, the borders of the lower 48 states were more-or-less set. While the United States had demonstrated the military and political will to claim this territory as its own, and the California Gold Rush of 1848 had already started the process of populating California, there was a LOT of land in between that was relatively barren of white settlement. French voyageurs and other fur traders had long plied the area, and the Mississippi was an important commercial waterway, but agrarian Anglo settlement was mostly lacking. And, the American Indians who lived on this land were still powerful militarily, politically, and culturally. After a detour for the Civil War from 1861-1865, the government was ready to start populating that area with white people. In series of Homestead Acts and related laws, they decided to give away the land to anyone who would "develop" it, ie institute agrarian settlement farming in the way that Euro-American people at the time understood to be "civilized." This was backed with a policy of aggressive military expansion into the Plains, providing a use for the newly professionalized military (especially a seasoned officer corps) that had just prosecuted the Civil War. When people speak of the genocide of American Indians, this is the last major and most deliberate phase of that process. They also gave away a lot of land to railroads and exterminated a lot of American Bison. The railroads gave the new farmers a way to bring crops to market, and the end of the Bison helped to starve the Indians.

So that was the "push." What about the "pull?" What convinced a person to migrate from the "civilized" East Coast to the West? First of all, many migrants came as family groups, unlike in the Gold Rush, when it was mostly single men (even those who had families tended to leave them behind, expecting to return home in a year or two). And they were pushed by a combination of population pressure and the decline of the farmland of the East. In general, New England and New York contain relatively poor farmland, with lots of rocks and relatively uneven terrain. By the middle of the nineteenth century, it was also starting to be exhausted. Further, generations of inheritances had left farms relatively small, unable to support the children of the current generation. If one sibling (usually the oldest brother) inherited the family farm, that left subsequent brothers (and of course all the sisters) to find something else to do to support themselves. There were a few professional occupations one could enter in a traditional small town, like the ministry. Others went to the cities, becoming clerks in mercantile houses, hoping to make enough money at business that they could then retire to a country estate (or at least the suburbs). Girls went to work in factories while hoping to meet a husband. Earlier in the nineteenth century people migrated to the Old Northwest, territory that had come to the U.S. after the Revolution (think Indiana, Ohio, Illinois) and to parts of the Louisiana Purchase (Alabama, Mississippi, Western George - this is what wrought the Trail of Tears). But by the Civil War, these places were filling up, too. So, people from these areas as well as from the East Coast struck out further West, taking advantage of the Homestead Acts and other inducements by the government.

As for outfitting oneself for the journey, there were a great many companies that sprung up to equip people for the journey, usually at the ends of the railroads in Illinois and Missouri (Independence, Missouri was by far the most common). A migrant would travel there with his family and purchase a wagon, pack animals, supplies, and a stock of seeds and other materials to help him once he established his homestead. He would then need to secure himself a place in a wagon train, an association of migrants all heading to the same place who banded together for mutual protection and assistance. Large and well-funded trains could hire a professional guide, but mostly they operated on simple democracy, electing a captain who had relatively limited authority. At night they would form a circle with the wagons to keep the wind out and the livestock in. People were free to join and leave trains at will over the months-long journey, dropping out due to an illness, or to move along a spur trail (say to stop in Colorado instead of continuing on to California or Oregon), or because they weren't getting along with someone in the train and wanted to try out a different one.

Once a person arrived at a claim, assuming they lived, the goal was to improve it, as "improvement" was the mark of civilization, and thus of a successful claim. Since wood was relatively scarce on the Plains, people usually started with a sod house. Literally, they would cut chunks of the turf and pile them up into a basic structure. They would also plant crops, though they usually needed to find a source of water for irrigation to make that possible. The Great Plains were sometimes called the Great American Desert, so irrigation was a priority. As more people came into an area, there would be neighbors for trade and socialization, merchants for comforts and necessities, and, eventually, the railroad. One there was a local railroad, homesteaders could take advantage of mail-order companies to order everything from the new steam-powered combines to houses to seeds (Sears is the most famous of these). Before the combine, animal and human power had to suffice for most tasks.

I think I've mostly covered the standard narrative. I'm happy to try my hand at follow-ups, of course.

Sources:

Howe, What Hath God Wrought

McPherson, The Battle Cry of Freedom

Unruh, The Plains Across: Overland Migrants and the Trans-Mississippi West

Limerick, Patricia, The Legacy of Conquest: The Unbroken Past of the American West

White, Richard, "It's Your Misfortune and None of My Own": A History of the American West

---. Railroaded: The Transcontinentals and the Making of Modern America.