Of course, we really have no firsthand accounts of first contact between Natives and Europeans from a Native perspectives. Much of what we know taken from European accounts, and in some cases from Indian art and iconography. Certainly, the fact that contact with Europeans was accompanied by staggering human destruction due to disease did not pass unobserved by Natives, so it is likely that the experience was in many cases highly traumatic.
What historians have been able to show is that most Natives had heard at least anecdotal accounts of European contact before it actually occurred. Some historians have suggested that New England Indians, for example, knew stories of Europeans from pervious contact with English by Algonquian-speaking peoples in Virginia. Similarly, John Smith's journals mention that Virginia Indians had heard accounts of English people in the Roanoke settlement, and may have even had personal contact with them. And European goods were not unknown to many Indian peoples, even long before they actually saw a white person. So Indians were not unaware of Europeans, at least not to the extent that many now imagine.
As for how they responded, most Indian peoples acted to quickly incorporate Europeans into their religious, political, and especially diplomatic and political lives. They sought alliances with them, wished to trade with them, and often tried to use them as a counterweight in their own complex relationships with other polities.
One very famous example illustrates the point. The Wampanoags, the people who famously attended the "first Thanksgiving" at Plymouth, had behaved charitably toward the Pilgrims because they hoped to make allies out of them. The Pilgrims needed friends to survive, and so did the Wampanoags, whose numbers had recently been decimated by a smallpox epidemic, and who faced destruction by their enemies in the area. So Massasoit, the Wampanoag leader, made a rational decision to seek alliance with the Pilgrims. A similar situation had prevailed in Jamestown, where Powhatan sought to incorporate the English settlers into regional politics.