Were the people who moved west between 1800 and 1820 mostly from the eastern cities moving west to start farms or were they people who had lived as farmers in the western parts of the original thirteen colonies?
The answer to this question largely depends on which part of the West one is referring to. A greater proportion of farmers that moved into the old Northwest (today's Indiana, Illinois, and Ohio) were from cities than those who moved into areas in Tennessee, western Georgia, and elsewhere. By and large, however, settlers in new frontier lands were farmers who had either been tenants or owned smaller plots of land in the east. They left for cheaper lands and generally for increased opportunity away from the elites who dominated most of the original thirteen states. In the North, many of these people tended to be immigrants, often from Germany.
The post Revolutionary War era saw multiple farmers' uprisings in protest against unfair taxation and currency policies enacted by self-serving elites, and many farmers, seeing the futility of reform in their own states, simply moved west to new lands. In both the North and the South, this process involved the forced removal of Native American people. This resulted in a series of very bloody wars in the Northwest. In the South, it also involved the expansion of slavery, especially after the cotton gin incentivized the cultivation of cotton.
It is also important to remember that many western lands had been purchased en masse by speculators, who then cashed in when the lands became open for settlement. In fact, by the 1820s, many lands in the Southwest were owned not by small farmers, but by large landowners, many of whom still lived on the east coast, and operated absentee plantations.