There were many clashes between the Native Americans and early European settlers; though much of it ended up in "mezcla" - a mix. For example, Europeans brought their Christian/Catholic beliefs, and as a result of European attempts to convert Natives to their beliefs, Natives (with the help of missionaries) combined many Catholic beliefs with their own, turning their "gods" into Catholic Saints. This, actually, is where we get many of the Saints that you'll hear about in Mexico and South America that Northern Catholics aren't familiar with (Our Lady of Guadalupe, for example).
Property ownership was different, as well. In Native American communities, an individual rarely owned something, especially land. They more so owned the right to live on that land, but it didn't mean the rest of the community had to stay away. So, if a Native sold, for example, some riverfront property to a settler, the settler might be shocked to see several people fishing off his lawn the next day. The Natives didn't understand the problem - how could anyone own a piece of nature? The Europeans, on the other hand, felt Natives had improperly taken back what they'd sold and gone back on their word - hence the nasty term "Indian Giver."
Another difference was the matriarchal versus patriarchal households and communities. Europeans normally had men in charge of families and communities, whereas Natives often had women running the show.
Basically, settlers (and later Americans) did what they could to "civilize" the Natives, trying to convert their "heathen" ways to European standards. This, of course, led to much conflict, unfortunately with Natives often losing their lives or being evicted from their own lands.
The Columbian Exchange had many impacts on the cultures of both the Native Americans and the Europeans. Different impacts were felt in different parts of the Americas. A few of these impacts came as a direct result of cultural exchanges while others came as a result of more material exchanges.
In many cases, Native Americans’ and European settlers’ cultures were affected by material exchanges between the two groups. One example of this was the change in some North American Indians’ cultures when the Europeans introduced horses to the New World. Another example of this was the change in the food cultures of the European settlers when they were introduced to foods that were native to the Americas. A final example of this is how smoking tobacco became an important part of many European settlers’ cultures when they were exposed to that plant in the New World.
In some cases, there were actual direct cultural exchanges. This can be most easily seen in places like Mexico where the European settlers actually mingled with the Native Americans. Much of Mexico’s culture took on a mestizo nature as European and native cultures mixed. Religion is, perhaps, the best example of this. The Native Americans adopted (or were forced to adopt) Christianity as their religion, thus changing their culture in fundamental ways. At the same time, however, aspects of their culture influenced Christianity as it was practiced in that country. For example, the observance of Dia de los Muertos (Day of the Dead) is a practice that came out of Aztec traditions.
In these ways, the Native Americans and the European settlers affected one another’s cultures.