The first major challenge that the Puritan colonists faced might be called an environmental challenge. They had to adapt to trying to make a living in a new land with an unfamiliar climate.
The original Puritan colonists' first challenge was the fact that they arrived in Massachusetts in December. This meant that they had to live off ships stores and help from the Indians until they could (again with help from the Indians) grow crops and harvest them. Almost half the colonists died of exposure and disease in that first year.
A second challenge faced was that of dealing with the Native Americans. Although Squanto and the Wampanoags were instrumental in allowing the Puritans to survive, the Puritans soon got caught up in conflicts between various tribes. This led to such conflicts as the Pequot War.