E. Earle Ellis' 1957 Paul's Use of the Old Testament (reprinted in 2003) has been another clarifying book recently in helping me put the whole Bible together; here are two good statements.
A swirling debate in recent decades is the degree of continuity between Paul's interpretive methods and those of his Jewish contemporaries (pesher midrash etc). Ellis gets to the heart of the matter well. After noting some resemblance between the two, he ultimately concludes:
The Pauline use of the OT cannot really be understood in terms of his Jewish contemporaries. . . . The affinities which occur are in peripheral areas and never reach to the heart of his thought. After his conversion the OT became a new book for Paul; all that went before now stood only as a prelude--a prelude set quite apart from all that was to follow. Although echoes of the preclude remain, the real meaning which the OT has for him lies at a different source. (83-84).
Ellis later expands on what he means.
Paul's devotion to Scripture was not that of a rabbi; he did not cite the Scriptures from a sense of duty or a love of theology or tradition, but because of their witness to Christ. For Paul, Christ was not only a factor giving added meaning to the OT but the only means whereby the OT could be rightly understood; it was not merely that he saw Christ in the OT but that he viewed the whole scope of OT prophecy and history from the standpoint of the Messianic Age in which the OT stood open, fulfilled in Jesus Christ and in His new Creation. (115-16; the second sentence of this quote was worth the reading of the entire book.)