Wow, so this is a deeply complex question. I'm not sure I can tackle all of it, but I can provide some context.

The debate on the nature and treatment of Native Americans began shortly after contact. The excesses of cruelty in the Caribbean and initial conquests on the mainland initiated an ongoing debate throughout the Spanish Empire on the need to protect indigenous populations, even if only so they could be around for future exploitation. For some like Bartolome de las Casas the transition from "savage" to "wronged people" began in 1515. de las Casas originally supported the oppressive use of Indian labor, but after witnessing the massacres accompanying the conquest of Cuba he dedicated much of his life to lobbying the Crown for an end to Indian slavery and the encomienda system. Charles V issued the "New Laws of the Indies for the Good Treatment and Preservation of the Indians" in 1542 as a way to prevent further Native American enslavement and tried to limit the power of encomenderos. Unfortunately, the New Laws were unable to completely reform practices throughout the Empire, but they testify to a colonial culture, at least in part, wrestling with the justification of cruelty toward Native American populations.

Jumping ahead in time to colonial New England, we see English colonists wrestling with similar issues in the wake of the Pequot War. Between 1634 and 1638 Massachusetts Bay Colony launched an assault on the Pequot Nation that would in many ways serve as the foundation for future English/U.S. Indian policies to come. The Pequots not killed in the conflict were hunted and sold as slaves to Massachusetts families or exported to the plantations in Bermuda and Barbados. Newell, in Brethren by Nature: New England Indians, Colonists, and the Origins of American Slavery, discusses how even though Massachusetts became the first state to legalize slavery in 1641, the colony adopted a less rigid, less race-based form of slavery that would later predominate in the southern colonies. In part this nuanced view of enslavement arose from the close, constant contact between Native American slaves and the colonial families they served. Depending on social connections and extended kin networks Native American slaves in Massachusetts could sue for their freedom, indicating a legal system that acknowledged their rights as more than a "savage".

However, these are simply preludes to the heart of your question. I included them as a means of showing a persistent sense of unease among colonial powers with regards to their treatment of Native Americans, even from the start of colonization. Warfare, territorial displacement, resource restriction, and all manner of violence continued against Native Americans for the next three centuries, but at least some people felt kinda bad about that.

Sadly, the overall transition from "savage" to "wronged people" occurred ridiculously late in U.S. history, and in many ways we are still a culture wrestling with the legacy of centuries of structural violence toward Native Americans. After the massacre at Wounded Knee in December 1890 there were still those advocating for the total extermination of Indians. Native Americans still fought for, and regularly lost, claims to lands and resources throughout the 20th century. In 1953 the completed Garrison Dam on the Fort Berthold Reservation in North Dakota flooded a quarter of the reservation, and covered 90% of the best agricultural land, forcing the relocation of the Mandan, Hidatsa and Arikara living on site. The Air Force took 340,000 acres of the Pine Ridge Reservation for aerial gunnery and bombing practice during WWII. In 1979, despite efforts by Cherokees and allied environmentalists, Congress sanctioned the Tellico Dam on the Little Tennessee River. The waters covered Chota, the former Cherokee capital, and other vitally important archaeological sites that could inform our understanding of Cherokee history.

This period of Native American history is not my direct area of expertise, and this will be a woefully superficial treatment, but a shift in U.S. culture in the 1960s and 1970s associated with the Civil Rights movement influenced the popular perception of Native Americans. The American Indian Movement was founded in 1968 and the standoff at Wounded Knee in 1973 brought national media attention to their cause. By 1990 conditions changed such that the Native American Grave Protection and Repatriation Act was signed, and helped return thousands of human remains and cultural artifacts back to their homes. The US now has national parks and national monunuments dedicated to bringing the worst of our history into the national consciousness.

Not to say all is well now. We are still a culture wrestling with the legacy of colonialism and centuries of structural violence towards Native Americans. There is still tremendous poverty, disparity in health care access, and deeply ingrained prejudice influencing the lives of modern Native Americans. We aren't there yet, but perhaps we are closer to reconciling our shared past than at any other point in the history of our nation.

Edits: To less overtly break the 20 year rule.

Paging /u/anthropology_nerd, /u/TheAlaskan, /u/Reedstilt

It changes depending on the times, and among different people. There are traditionally two tropes for European views of Aboriginal peoples, and the popularity of them ebbs and flows along with other social trends. You've likely heard of them. One is the "Noble Savage", where the Aboriginals lived in an idyllic society free of war and conflict before the Europeans showed up, while the second trope views them as brutish savages that lived much worse before Europeans arrived. While modern (I can only talk until 1995) views are more nuanced, many modern views are a mixture of both.

I would say that the recognition by a majority of Americans that the treatment of Aboriginals is wrong is fairly recent, although polling on this issue is sketchy and it may even be that a majority of Americans do not believe that the Aboriginals were fundamentally wronged. But again, this is getting into modern politics, and also its important to realize that political views are on a spectrum, and it is quite possible to hold the view that the Aboriginals were savages while also believing that America's actions towards them were unjust. If you have any more specific questions or you are still confused feel free to answer, I know it is a complicated question.