A short history of the Talmud
Jewish Law has it's foundation in the books of the Torah: the first five books of the Bible. Although there are something like 613 commandments (mitzvot) in the Torah, there are many areas of life that are not directly addressed by the written law. And society has changed in ways that aren't explicitly accounted for by the text. For instance, Moses commanded:
Mark that the Lord has given you the sabbath; therefore He gives you two days’ food on the sixth day. Let everyone remain where he is: let no one leave his place on the seventh day.—Exodus 16:29 (NJPS)
Originally, that mean to not leave the camp to go and gather manna. Traditionally it meant not to leave the area of the city as defined by the city walls. (See Jeremiah 17:19-27.) But these days, we don't have well-defined cities. I live near Los Angeles, but unless you pay careful attention to the signs on the side of the road, you can't tell if you are in Burbank, Glendale, Pasadena, Los Angeles or unincorporated Los Angeles county. Am I breaking mitzvot by moving around the metropolis? (The Orthodox Judaism solution is to create an eruv.)
To answer practical questions, such as this, Rabbis began to formulate a system of interpretations based on their understanding of the Mosaic principles. According to tradition, this Oral Torah could be traced back from student to teacher to Moses himself. After the destruction of the Second Temple, Rabbis began to write down the Oral Law, which became the Talmud.
Jesus and the Talmud
As it happens, during Jesus' time there were (according to Josephus and other historians) three mutually dissenting views on what constituted the Jewish Law:
Sadducees, who rejected not only the rabbinical traditions, but all of the Tanakh outside of the Torah.
Pharisees, who shaped and promoted the Oral Torah in addition to the Tanakh.
Essenes, who seemed to have accepted not only the Tanakh, but a wide range of other texts. They seem not to have accepted the Oral Torah. (If the Qumran community was Essene, we could be more certain about their beliefs.)
Jesus (and the early Christian movement) mostly interacted with the Pharisees and theologically seemed to have the most in common with them. (See Matthew 22:23-33 and parallels, Acts 5:17-42, and Acts 23:1-11.) Some scholars speculate that John the Baptist was an Essene, but otherwise we have no record of Jesus interacting with that sect. As you point out, Christianity accepts the texts that the Pharisees used as their scripture. (The Tanakh was completed and in the process of being canonized, while the Talmud remained strictly oral at the time.)
But Jesus roundly rejected the Oral Torah as misguided:
Woe to you, scribes and Pharisees, hypocrites! For you tithe mint and dill and cumin, and have neglected the weightier matters of the law: justice and mercy and faithfulness. These you ought to have done, without neglecting the others. You blind guides, straining out a gnat and swallowing a camel!—Matthew 23:23-24 (ESV)
Paul interpreted Jesus' ministry as invalidating other Jewish practices:
But when Cephas came to Antioch, I opposed him to his face, because he stood condemned. For before certain men came from James, he was eating with the Gentiles; but when they came he drew back and separated himself, fearing the circumcision party. And the rest of the Jews acted hypocritically along with him, so that even Barnabas was led astray by their hypocrisy. But when I saw that their conduct was not in step with the truth of the gospel, I said to Cephas before them all, “If you, though a Jew, live like a Gentile and not like a Jew, how can you force the Gentiles to live like Jews?”—Galatians 2:11-14 (ESV)
The early church even rejected imposing certain laws written in the Torah, such as the kosher rules and circumcision, on gentile believers. As the church spread to Greek and Roman cities and beyond, the Jewish observance of the mitzvot also diminished. There is no evidence that Christianity ever advocated the Oral Torah which eventually became the Talmud.
Mason Wheeler summaried the Christian perspective on the the Talmud well. I will add that the Talmud provides a good deal of useful historical material for anyone trying to understand the origins of Christianity.