Why & how did the US government conquer the Native American nations?

Your question needs revision because there were relatively few Native American "Nations" and those were mostly in the Northeast where encounters were much earlier than the bison killing and removals mentioned in other answers. And some of the Nations made the choice to ally with the French in the wars against the British before the Revolution and thus before the "U.S. Government" existed.

There was pressure on the small tribes right from the beginning who often warred with each other but were devastated by disease even in the short time between the arrival of coastal fishermen and the Mayflower landing - taking over the site of a native village which had lost so many people it had merged with tribes further inland.

The first removals were actually in Georgia with the discovery of gold on land promised to tribes long before the west coast gold rush.

It is hardly fair to talk about the Plains Indians as Nations when they were not organized that way even when the Government pretended they were and signed treaties with people that were designated by the government and not representative of more than their own tribe.

If you want a view of the ruined condition of Native American life, both North and South America, you might find the book '1491' which discusses population levels and the effects on environment of disease destroying the people who used the land and the forests.

Why: The white settlers thought that the Native Americans lacked a concept of God and were therefore uncivilized. To boost the meatpacking industry, the white settlers killed large numbers of bison, effectively putting strain on the way of life enjoyed by the Plains Indians.

How: Wars against tribes, reservations. et cetera

Why: To gain land in the West and expand the nation.

How: By removing Indians from their lands and moving them to reservations. Violence was often used to move them and many were killed in the process.

Apaches never surrendered to the whiteman,, americans give a good thing and a bad thing to them,, a horse and firewater whisky.

Disease,starvation and guns.

Who says we're "conquered"? Please show me any tribe who signed any document of surrender. We're all still here.