How was religion connected to the government in the New England colonies?

The first colonists to New England, the Pilgrims (as most know), landed at modern day Boston. These where religious seperatists from the Church of England and established Plymouth as a 'city on a hill' as there religious utopia.

As a opposite example, Rhode Island was founded by people who disliked the above religious society, and was established with complete religious freedom. The same was established by William Penn and the quakers in Pennslyvania, and the catholics in Maryland.