Wethersfield's First Church of Christ is one of the few churches can claim a heritage extending back to the first part of the 17th century. Founded in 1635, the church's present building dates from 1761. The community has changed denominations several times, going from a formal Congregational church to a member of United Churches, before becoming an independent Congregational church. You can trace the entire history on their website.
The church has worked hard to make their history accessible. Besides that webpage, the website includes email addresses for their archivist and historian. Make sure to look at the top of the history page. The Connecticut Church Records Slip Index also includes records from First Church. They are detailed in an article posted on American Ancestors. Their work should make life easier for many of us.