Hebrews 1-13, James 1-5, First Peter 1-5, Second Peter 1-3

The Letter to the Hebrews is as it suggests an attempt to explain Christianity to the Jews, something Acts of the Apostles featured but the emerging faith started to reject as its mandate over time, so it's good to have a letter dedicated to it in the New Testament.

If you were ever wondering why Melchizedek appears to be so important to Christians despite only a cursory reference in Genesis, it's because of a generous emphasis in Hebrews.  Jesus, meanwhile, is described as the ultimate high priest, among other things.

Hebrews uses as part of its argument a list of Old Testament figures who were known for their exceptional faith, including Abel, Enoch, Noah, Abraham, Sarah, Isaac, Jacob, Joseph, Moses, Rahab, and words that to the effect mean "etc."  In that sense it rings true with material in the Old Testament itself.

Although at one time ascribed to Paul, this is no longer the case with Hebrews, although the link of Timothy is present.

The Letter of James is probably from the same ambiguous James from Acts of the Apostles who proved to be a hassle for Paul's ministry.  This is an appropriate follow-up to Hebrews, considering this James was very much interested in retaining the Jewish tradition into the emerging Christianity.

It is in fact written to Jewish converts.  It also has an exhortation to endure trials, and an emphasis on good acts but no partiality (except, y'know, the Jews).  James references the Golden Rule, although of course he says to keep the old rules, too.  Interestingly, a fig tree is referenced, the only time outside of the gospels where it is.  It apparently really was an important legacy of Jesus's.

First Letter of Peter is also a letter to converted Jews, this time addressed to those of the Dispersion.  Rome is referenced as Babylon (as it will be in Revelation as well).  Mark is referenced.  It's a letter concerning general rules of conduct.

The Second Letter of Peter assures the recipient that Jesus is not a myth.  It also, unfortunately, continues the biblical tradition of slandering Balaam (as does Revelation).  The second coming is referenced.