How did the apostles interpret the Hebrew Bible? There has been debate among Christian scholars. I personally think the evidence is very clear that the apostles did not slavishly interpret the Hebrew Bible literally nor did they even quote the text slavishly when citing entire verses. There are many passages that could be mentioned, but the classic example is Paul's deliberate misinterpretation of the word seed/offspring in Genesis 13:14-16 in Galatians 3:15-29. Paul knows that "offspring" in Genesis 13 is a collective noun and refers to more than one person. However, he felt free to deliberately misinterpret the text to make his point. One could also point to his allegory of Galatians 4:21-31 for another clear example that Paul (and other New Testament authors) were not stuck with a narrow range of uses of Scripture. Scripture gave them a resource to make a point that they felt the Holy Spirit wanted them to make. The point was more important than the literal wording or even the literal meaning of the text.
We see a similar phenomenon with Christian scribal activity in the first several centuries. Many Christian scribes felt free to make additions or modifications to the text. Many textual variants of course are scribal errors or were attempts to fix what later scribes believed were earlier scribal errors. However, there are a lot of modifications made to try to improve the comprehension of or the grammar of the text. If they believed the words of the text themselves were powerful one would not expect to see even these kinds of changes. However, some scribes went far beyond that adding, sometimes, even extensive material to conform scripture to their beliefs. Scripture was clearly important to these early scribes as an authoritative source of doctrine, but again, their behavior suggests they believed the Bible to be something other than verbally inspired otherwise you wouldn't have these kinds of changes.
My next and final point may seem a bit out of place given that I'm talking about Christian understandings of their Scriptures, but the earliest followers of Jesus were Jews so it's meaningful to know what beliefs they would have inherited. In the Vermes book I mentioned above, he discusses the role of the Dead Sea Scrolls in helping us determine the original text of the various writings of the Hebrew Bible. Three distinct textual traditions with significant variations in the text itself have come down to us; the Samaritan Bible, the Septuagint, and the Masoretic Text (Hebrew text). The readings in our Bibles largely (perhaps completely? I'm unsure) follow the latter two in some way shape or form. There are many minor variations between the texts (e.g., the Samaritan Bible often replaces references to Jerusalem with Samaria), but there also are some major differences. For example, the text of Jeremiah in our Bibles is mainly derived from the Masoretic text. However, the Septuagint text is about 1/7 shorter and is arranged in a different order. Which is original? Either, neither, or both? Who knows? Manuscripts reflecting both traditions were found among the Dead Sea Scrolls. For other texts, we have examples reflecting the traditions in the Samaritan Bible in the Dead Sea Scrolls. This means that during the time of Jesus and the early church there were a variety of texts that were, in some cases, quite different, and it wasn't imperative to adjudicate which version of the Scriptures was official and "God's words." Even beyond that, given the disarray we have in some cases, how confident are we really that our reconstructed Old Testament is that close to the wording of the Hebrew (or Aramaic) original? That alone seems to be a practical problem for those who hold to a doctrine of verbal inspiration (though of course not a refutation).
Some may say I'm being too modern by requiring literalism on the part of ancient Jews and Christians. I would say that is precisely my point. This doctrine of Scripture is very modern (in that it fits a modern Western worldview better than an ancient one), but I often fear that fact goes unrecognized. If one wants to maintain that Scripture is the word of God, one either needs to recognize that this belief is far more innovative than is typically admitted or find a way to affirm this belief that doesn't necessitate and isn't tied to a "literalistic" method of reading Scripture.
To get a lay of the land, you could read Three Views on the New Testament Use of the Old Testament.
 E.g., See the extended discussion by Enns in Ibid. pp. 180-85.
 Though I have not read it, I have heard The Orthodox Corruption of Scripture by Bart Ehrman covers this ground well.
 A classic example is the Johannine comma an addition to 1 John 5:7-8 (compare KJV with NRSV), though many other texts could be pointed to. Wikipedia has a lengthy discussion of this passage that could serve as a starting point. Note their list of other disputed passages.
 Vermes does not discuss my example here of Jeremiah but discusses the diversity of textual traditions found at Qumran on pp. 213-14.