Each European faction had different motivations to claim vast stretches of the continent, the Spanish for instance, claimed to do it in the name of God and the Catholic Church, the considered their given right and obligation to evangelize the native american population. England wanted more land and spices to trade and work with, France wanted pretty much the same as England.
The three of them interacted both peacefully and violently with the native cultures, in its broadest terms, the Spanish preferred to mix with the native Aztec, Maya and Inca populations, they were baptized and several times became part of new native-spanish (mestizo) families. Both England and France interacted with the natives in a more capitalistic way, either to trade with them peacefully, or settled on their land and either ignore them or fight them out.
