In Lesley Hazelton's biography of Muhammad, The First Muslim, she discusses an ascetic named Bahira that Muhammad encountered while a camel boy in his youth, who was known to have a book that may have been a Bible, but it is also mentioned that he likely would have been written off by the more respectable members of Muhammad's group as a madman. This could have been his first encounter with the Bible, if he saw it. Books were rather rare in the time period, and this helped the 'People of the Book' to differentiate themselves from other, oral traditions.

As others have said, Muhammad was illiterate (this is a point during his night on Mount Hira when the Quran is revealed in some versions-Gabriel tells him to 'recite,' but he can't read). Muhammad would have had some experience with both Christianity and Judaism, but it would have been more of an oral introduction due to his travels in trade caravans, and the myths told would be adapted to the Middle East. Many of the debates going on at the time period had to do with hair-splitting issues of Christianity that were irrelevant to people like those in the Middle East: was Jesus divinity embodied in man, half-man and half-divine, or entirely man?

As Hazelton recounts it, Muhammad's goal was to create a religion that would help peel back Mecca's increasingly capitalistic values and return to the tribal values that had come before. Karen Armstrong in A History of God explores a similar thesis: within the previous two generations, Muhammad's tribe, the Qureysh, had become very wealthy and many of the communal values that had until then kept the tribe alive were beginning to erode. As an orphan (his father, mother, and grandfather all die while he is very young), he was particularly sensitive to these changes.

His understanding both of Christianity and Judaism shows that he was not familiar with them as we understand them: for example, he hoped that Jews would accept him as a prophet, which is at least part of why the Biblical story of Abraham sacrificing Isaac is modified to Ishmael, because the Arabians were believed to be descended from Ishmael. He was surprised when he was not accepted by Jewish people as a prophet; I believe he was also under the impression that Jesus had been accepted as a Jewish prophet.

As for Yathrib (Medina): following the failure of the Jewish tribes to adopt Islam, they were systematically driven out of Medina. First, the Qaynuqa, after a claim that they had nailed a young, veiled Islamic woman's skirt to expose her when she stood up. (Hazelton points out that at the time of the event, veils had not been adopted for Islamic women and, during Muhammad's lifetime, were intended only for his wives).

After a dispute with a Beduin tribe, Muhammad tries to get the Nadir Jewish tribe to foot the majority of the bill, which they balk at; he has their date trees cut down and drives them out of Medina, though they leave on a celebratory note, not as outcasts like the Qaynuqa. Finally, after considering the Meccan leaders' (Mecca had driven Muhammad out) positions during the Battle of the Trench, Muhammad has the final tribe, the Qureyz, eliminated: all men old enough to shave are beheaded and the majority of women and children are sold into slavery.

Another tradition you might find interesting in terms of shaping Muhammad's worldview is Manichaeism, an earlier syncretistic religion founded in Iran. The founder, Mani, even went so far as to use the term 'Seal of the Prophets,' and this was ~400 years prior to the arrival of Muhammad on the scene. They were heavily persecuted and their pacifism in the face of this more or less ensured they wouldn't last, but placing themselves as the 'seal of the prophets' is pretty interesting: while Mani claimed descent from Zoroaster, Buddha, and Jesus, Muhammad claimed his prophetic tradition from Abraham and Jesus.

When you say early texts, what do you mean? The current Biblical canon was pretty much put together in the 3rd-4th centuries (and most of it was already settled by the end of the 2nd century). So, if he had read Christian or Jewish texts (which is possible: there were Christians and Jews in 7th century Arabia) they would likely be rather similar to the Greek texts you can read today. Are you referring to "early" in the sense of written early? Or about early things (like Genesis)?

He may have, possibly. We do know that, when he sought refuge in Yathrib (Medina), he encountered various quarreling tribes in the town, including pagans and Jews. His message would soon be accepted by the people of the town, and the tribal infighting would come to an end. Although I haven't read an explicit reference to Mohammad reading Jewish holy texts, it is pretty likely that he had contact with the Jews of Medina, from whom he could have gained insight into the Torah (if he did not already know its contents before). Since there were also Christians in western Arabia (and because Mohammad had fled to the Christian kingdom of Ethiopia in the First Hegira) he may well have had a similar knowledge of the Bible.

Source: Esposito, "The Oxford History of Islam"

P.S. Are you asking this question because ultimately you want to know how much Judeo-Christian beliefs had influenced Islamic roots?