In the comments below, Laura rightfully asks whether or not knowing the 39 articles are essential. I’m sympathetic – I have a similar reaction when people parade Saint Hooker as the theological answer to our polity problems.
Perhaps there is another question: is there a historical and theological narrative to being Episcopalian we assume people should share? Is there a taxonomy or a lexicon of dates, facts, events and perspectives that we expect to be normative? There may not be. Although I understand the sentiment against idolizing the past, or claiming it uncritically, I wonder if our reticence toward naming our tradition inhibits us in other ways.
Are there some basic ideas we expect people to know about our tradition? I think, for example, that new Episcopalian should be able to distinguish why we are organized differently than congregational churches, peace churches, and The Catholic church.
But should we be affirming denominationalism? Most of the time I’m anti-denominational, but on the other hand, there are aspects of Episcopal culture (beyond gin and tonics), such as its intellectual and musical heritage, that would want to pass on. The organizational potential of the Episcopacy has yet to be tapped.
I don’t think that such a list would be long. But there may be a pedagogical issue here: I’m inclined toward memorizing, drilling, and testing as legitimate (but not comprehensive or complete) aspects of learning. How do we describe the shape of being Episcopalian, and what are the events or facts that articulate that shape?
Another way to look at it is we’re playing “connect the dots” with our denominational heritage or ethos. What are the dots? Do we need Hooker or the 39 articles?
This is separate from what may be essential for Christians to understand. Some could be minimalist: merely be able to be a friend of God and others. Others might require adhering to dispensational theology. I might leave it at believing that the church’s teaching about Jesus’ resurrection is the location for our holiness. That is for another blog.
The Divine Latitude 
I’m even now procrastinating writing a lesson about Church History, part of which is trying to find a way for people to explore why their church is like it is. I found that there were three questions (with some subquestions) that helped define a denomination:
1) How do we make decisions as a church?
2) How do we choose who should preach?
3) What do people have to do to become members?
If you know the answer to those questions, you know quite a bit about your denomination. None of which are specifically about its history or its doctrines, but the history and doctrines are built into that. I suspect very few Episcopalians have thought about church in this way.
For me, one of the things that makes learning Church history so valuable is so that you realize that things haven’t always been that way. If you’re talking about things to memorize, one thing I wish every Episcopalian had to know by heart is the first phrase of the first sentence of the preface of the 1549 BCP:
There was never any thing by the wit of man so well devised, or so sure established, which in continuance of time hath not been corrupted.
Words for the church to live by, I think.