To establish a "city upon a hill": The puritans believed that they had a special mission to create a perfect Christian society in the New World. They wanted to establish a "city upon a hill," a model for all other Christian communities.
To convert Native Americans: The puritans also wanted to convert Native Americans to Christianity. They believed that it was their duty to spread the gospel to all peoples.
For economic reasons: The puritans were also motivated by economic reasons to settle the New World. They wanted to find a land where they could prosper and have a better economic future.