Since I finished reading through my Greek New Testament nearly a month ago, I've been working through the Apostolic Fathers in fairly unsystematic fashion. I've managed to acquire a number of helpful resources: For those with only a tenuous grasp of Greek, I've found Rodney A. Whitacre's A Patristic Greek Reader (Peabody, MA: Hendrickson, 2007) a very helpful introduction to the Greek texts of early Christian writings. For the complete text of the Apostolic Fathers I've been reading Michael W. Holmes's The Apostolic Fathers: Greek Texts and English Translations (third edition; Grand Rapids: Baker Academic, 2007). I also have the first volume of Bart Ehrman's revised Loeb Classical Library edition, The Apostolic Fathers (two vols.; Cambridge: Harvard University Press, 2003 [volume 1; volume 2]), but I have stuck mostly to Holmes's one-volume edition.
In the last month I've read the Didache, 1 Clement, and Ignatius' epistle To The Romans. What has struck me thus far is the different ethos with respect to biblical tradition that characterizes those three texts. The Didache (an early church manual with strong resonances with Matthew) sounds a lot like a NT text, though considerably more focused on the life of the church. As a NT scholar, I'm used to inferring the church's conduct from gospel texts; the Didache, even more than the NT epistolary texts, explicitly addresses how the church ought to pray, celebrate the Eucharist, welcome itinerant missionaries, etc.
1 Clement, on the other hand, reads a bit more like a Pauline epistle (with considerable differences, to be sure). But 1 Clement, like the NT texts, lives in a world defined by and filled with Hebrew biblical traditions. Here Abraham, Isaac, Jacob, Joseph, Moses, David, Job, and others appear not just as characters but also reference points for apprehending and interpreting late-first-century realities and responding appropriately to them. In 1 Clement the church continues to find nourishment and succor from her Israelite roots.
Ignatius, however, strikes me differently. Granted, I've still six Ignatian letters to read through, and I don't know what I'll encounter there. But in To The Romans, Ignatius doesn't take advantage of the opportunities afforded by Hebrew biblical traditions to make sense of and respond to his impending martyrdom. He clearly does understand what will happen to him, but Israel's story doesn't seem to play much, if any, role in how he understands it. I'm wondering, in other words, if Ignatius, unlike the author of the Didache, Clement, or the NT authors, distinguishes between the stories of Christ and of Israel.