How did contact between the European arrivals and the native peoples of the Americas affect both groups?
The above answer is an excellent one, and it is true that any single answer on a site like this is only scratching the surface. That said, there are several small "impacts" that can be gathered under one or two larger umbrellas. It can be argued that the colonization of the Americas led to both a diversifying and narrowing of experiences and worldviews for both the Native Americans and the Europeans. In other words, they both learned and lost learning.
The Natives did indeed gain from the Europeans bringing items from the "old world" over. For example, they gained access to horses and livestock, and learned relatively modern animal farming practices, including the concept of dairy farming and growing wheat. They were introduced to cheap paper-making and a more simplified writing system than most tribes practiced. They were also introduced to cultures completely different from their own and gained a greater breadth and depth of knowledge by being taught about the "old world" and the other countries and cultures the Europeans had already made contact with, such as the Far East.
On the European side, they benefited greatly from learning about the foods the Natives ate, particularly corn and sweet potatoes. They learned new ways to take care of their animals using American products and often had great success in that area. As the answer above notes, they also learned valuable lessons such as surviving in winter and thriving in their new environment. Additionally, they similarly gained knowledge by being exposed to the wide variety of Native tribes and their customs. Both sides gained new languages, as well.
On the other hand, as the European numbers and influence grew, all of these experiences began to be lost on both sides. Both sides gained a new source of conflict, but one that cut each off to what the other had to offer. Because the Natives lost more lives and land in this conflict, they were the ones to suffer the effects of this far more. As the Europeans took over their lands, they also pushed their values on the Natives, making them more culturally as well as physically colonized. This also led to a "flattening" of the understanding of Native culture amongst Europeans that has lasted to this day. Originally, the tribes of the Americas were very diverse. In reality, they still are, but as the Europeans eschewed their culture they began to reduce all tribes to one single "Native American" society and dismiss the valuable lessons the Natives had to offer. While the Natives suffered far more, the Europeans also suffered an ultimate loss of cultural depth and breadth, as well as an ally, due to their actions.
This contact affect both groups in so many ways that we can only scratch the surface in an answer on this site. Let us look at one good and bad impact for each group.
The Native Americans were impacted negatively in that their land was taken from them and huge numbers of them died from the Europeans' diseases. On the other hand, for as long as they remained free, many Native peoples benefitted from European animals like horses and from European technology.
The Europeans were impacted positively in that they gained many things even outside of the land. For example, Native Americans helped Europeans to survive the early years and helped them learn how to thrive in the new lands. On the other hand, contact with the Natives gave many Europeans opportunities to give in to the darker impulses. It allowed them to treat other human beings as if they did not matter and as if they only existed to be used by the Europeans.