I notice that this question asks about “European colonization of the New World.” That implies that we are talking about all of the New World and not just the British colonization in what is now the United States. My answer will touch on native peoples in various parts of the Americas, not just on those in what is now the United States.
Perhaps the biggest impact of European colonization on natives’ lives is that it forced natives to live their lives without many of their compatriots. European colonization brought diseases that were deadly to the natives. Some scholars estimate that as many as 95% of all natives were killed by these diseases. This would have meant that various communities and societies were devastated, leaving the survivors to deal with life in a situation where so many of their family and friends had died.
A second impact is that European colonization destroyed the native religions of many people, particularly in Spanish America. The Spanish tried hard to impose their religion on the native people. This helped bring about a situation where the natives largely abandoned their indigenous beliefs and became Christian (though they did synthesize their beliefs with Catholic beliefs to some degree).
A final impact is the impact of technology and horses. When the Europeans came to the New World, their technology (metal knives, cooking pots, woven cloth, etc.) became part of the native economy. In some instances, as with the Plains Indians, horses completely changed the natives’ way of life. Native Americans were now using technologies that changed the way they went about their daily lives.