In the mid-nineteenth century, white Americans wanted to assimilate the Indigenous people to white customs and essentially force them to abandon their traditional beliefs, values, and culture. They believed that it was their right to take the Indigenous people's land and their mission to Americanize the "primitive" Indigenous people and make them behave like "civilized" white people; the main method used to assimilate the Indigenous people was education.
The political structure of that era came up with the idea of opening boarding schools in which Indigenous children were taught to speak the English language, given English names, styled to look like white people, and asked to behave like white Americans. The were also basically forced to convert to Christianity, as Christians (both Catholics and Protestants) believed that they could only be "saved" if they accepted Christianity and abandoned their spiritual beliefs and traditions.
Many Indigenous people were forced to accept these new "educational" and religious programs, but many also refused to participate in them and continued to teach their children about their culture. Many also refused to give up their land and assimilate to white culture, which culminated with many conflicts between the Indigenous people and the white settlers. These conflicts are considered to be a part of the long-lasting American Indian Wars.
In 1887, the government issued the Dawes Severalty Act to assimilate Indigenous people to white ways of life and to enable white Americans to gain more land and control over the resources (such as gold) that were on that land. This was achieved by breaking up the tribal lands, which meant that Indigenous people were allowed to own land only individually and not communally. The land that was given to separate Indigenous families to cultivate was actually quite bad and very small, and many weren't even able to grow anything on it, which meant that they lived in great poverty. The tribal land that was left was distributed to white settlers for farming or industrial purposes (such as building railroads).
Despite the Americanization of the Indigenous population, many were still discriminated against. They were forced to give up their culture, and even when they did, they still weren't well accepted into the American society. Thus, the attempts of the white Americans to influence, or rather assimilate, the Indigenous population to white culture had dire consequences.