The earliest American writings come from which three groups?

I am not sure what exactly you mean by "groups" but I can
offer the following suggestions:

1.  One group of early
American writers where the actual pilgrims who arrived on the Mayflower from England. 
John Bradford would be the most widely known.  His writing recounts the events of the
passage over the Atlantic as well as his and the others' first impressions of the New
World.  Through his writing we can see the Puritan view of the world, his understanding
of God, and the hope he and the others have for this new
land.

2.  Another type of early writing are the sermons of
the preachers of the day.  One of the most famous published sermons is "Sinners in the
Hands of an Angry God" by Jonathan Edwards.  Through these types of readings we can see
the Puritan understanding of the God and their relationship with Him.  This sermon
certainly illustrates the reason Puritans were considered "God-fearing"
people.

3.  A third type of writing could be the poetry of
Anne Bradstreet, a Puritan woman writing in the 1670's.   For a female poet to writing
and published in her lifetime is interesting to begin with, but her poetry has stood the
test of time because she captures, with heartfelt emotion, the feelings she has for her
family and her role as in the family and the community.