What kind of people headed west between 1776 and 1850?

Well, in the case of immigration to the U. S. around 1776, then there was very little.

However, talking about citizens moving West in the U. S. in 1800 - 1850, you had a number of people move West after the Louisiana Purchase in 1803, specifically those who wanted to explore new land. Furthermore, in 1848, the California Gold Rush prompted many people to move West to California and the nearby areas to look for gold and "get rich".