Why did Americans begin moving west in the 1800's?

The simplest answer would be "Profit." The U.S. Congress said that "the Native Americans stand as guardians to the treasure vaults of this nation." They were referring to the lands held by the Native Americans, to the west of the land that the U.S. owned.

Western expansion was for the sole purpose of profit, personal as well as the profit of the U.S. Natural Resources of gold, silver, agriculture and more-so to claim these lands for the U.S. before anyone else could claim them (Spanish / Mexico). Everyone, except the Native Americans, stood to profit from claiming these lands; by any means possible.