I should add that Congregationalists in the US are the modern descendants of the Puritans and Separatists who came to America. I would consider them more Calvinist in their theology than the actual Puritans were, however.
The problem with the puritan movement is that it never successfully defined what it was for. You had a vast array of theologies within the movement, with congregationalists vs. presbyterians, antinomians, baptists, Fifth Monarchy types, up into the anarchic Levelers. The only thing which united them (and why we have the one category, without them having any common theology), was the one thing they were against: the Established Church of England. Even the early puritans who nominally conformed to CofE doctrines in the 1580s-1590s, were essentially working the 'outside inside' strategy of trying to alter the CofE theology by staying in the church, saying all the right things, and rise up through the ranks over time. But their tactic was stalled by the turn of the century: the leading Anglican Divines (Bancroft, Bilson, Hooker, Andrewes), adamantly showed that these people are not our friends. The Puritans were starting to advocate for divorce and remarriage, wanted to rewrite the creeds; many of them wanted to abolish the episcopacy, the vestments, the liturgy, and the sacraments; they introduced 'double predestination', and 'precisionism' in moral theology. Ames, and Perkins, and the Plain Catechism, and the Golden Chain, these were pretty destructive innovations (as perceived by the orthodox side). By the 1620s the Puritans were increasingly on the outside of the Established Church, so the tactic switched to rile the general populace in a revolt and revolution. They succeeded; a heinous revolution was raised, and the entire country was overthrown. The Anglican Articles and Liturgy were thrown out. In the 1640s and 1650s the whole gamut of these people burst out into the English national scene. The Levelers advocated for communism (in the 1650s!). The adamists began to walk around naked on the streets. Even the more 'mainstream' Presbyterians never managed to gain wide acceptance; John Owen was a congregationalist and a baptist, yet they still consider him one of their great thinkers, having so few of their own. And that doesn't even cover the mess which was the American puritans, which are its own separate category. By 1660, there was the glorious Restoration of the Church of England; the bishops were back, and quickly set to work of restoring orthodoxy back in England. Every single priest and rector of a parish was required to subscribe to the Articles and the Prayerbook. Those who insisted on clinging to their errors were thrown out, as a literal danger both to the Church and even to civil society. In 1666 there was the "Great Ejection" where something like 6000 puritan rectors (a huge growth from the years of the 'Puritan Republic') who refused to abide by the Anglican doctrine and theology were thrown out, and ejected out of holy orders. The 1662 BCP was finally codified and passed as well. By the 1680s, the Puritan movement was essentially over. You had a few scattered 'dissenters' (non CofE ministers and churches) throughout England through the 1700s. The Quakers (which were considered heretics) came directly out of the Puritan movement in the 1660s, but they had more purchase in the American colonies than back at home. King Charles II granted William Penn a vast swathe of land in the New World, and he took most of his Quaker brethren there, to found what came to be called by his name, Pennsylvania.
I would probably be a bit further on the Catholic side than middle of the way. I am not anglo catholic but I am a high churchman who really likes the East also. I place heavy emphasis on the Church Fathers.
Does the Westminster Confession along with the Larger and Shorter Catechisms give a good perspective on the Puritan outlook, at least as it was understood during the Interregnum? That's what I was told.
Some Puritans were similar to Presbyterians in this respect. As I said before, there was a great diversity in theology among Puritans. They tended to be more pragmatic and focused on living as Christians; they were less interested in formulating or affirming various creeds and confessions. I imagine that they wouldn't have found much in the Westminster confession to object to. Apart from the form of church government, their basic doctrine tracked Presbyterianism pretty closely.
The preface to Hooker's Laws of Ecclesiastical Polity has a very fascinating description of the Puritans. It's too long to quote, but to summarize, he describes them as convinced they have the true understanding of scripture, in contrast to the historic Christian faith, and impervious to logic. And their core belief was that society needed to be transformed into a utopia by removing the king and the bishops. This was, of course, eventually happened except for the utopia bit. In The New Science of Politics, Eric Voegelin identified the Puritans as heirs to the same Gnostic legacy as the Marxists and the Nazi regime (from which Voegelin was a refugee): the belief that utopia will be achieved in this life, coupled with the belief that the government rather than the Church is God's representative on earth. Based on Ananias description, it seems like the term Puritanism was also applied to pietism, so I assume Hooker must have been speaking about a specific subgroup.
I've often said that Puritans are a perfect example of why you shouldn't allow yourself to be defined by your enemies. I'm more familiar with American Puritanism than with English, obviously; a lot of the hostility trad Anglicans have for Puritans goes back to the English Civil War and Cromwell. It's not (exactly) a religious beef. Puritans didn't have a distinctive theology, really. Apart from Church government, and broadly speaking, there's not much that separates their theology from Knoxian Presbyterianism. It was more their style of life and approach to living out Scripture that makes them interesting to me. (Plus I love the sermons of Jonathan Edwards.) American Puritanism is quite different from the English strain. Again, much of the calumny against the Puritans came far later in the 1800's (e.g., "The Scarlet Letter") and didn't particularly accord with anything the actual Puritans said or did (granting that Cotton Mather was a real jerk and did the Puritans no favors). I often say I'd have been a happy Puritan in former times, but I doubt they'd ever have accepted me: my liking for the BCP and episcopal government would probably have doomed me from the start, however close we are in actual theology.