Unfortunately, aside from the New Testament, we do not have many early sources on the teachings or even the person of Jesus. Josephus briefly mentions Jesus in his late first-century work Antiquities of the Jews, and the writings of Tacitus and (probably) Suetonius also mention Jesus, none of the three historians, however, record Jesus’ teachings. Josephus and Tacitus provide basic biographical information on Jesus (notably, his crucifixion ordered by Pontius Pilate) while Suetonius mentions a certain Chrestus once in his Twelve Caesars, whose followers rebelled against the Roman Empire, leading to the expulsion of Jews from Rome by Emperor Claudius. Aside from the name Chrestus bearing similarities to Christ or Christus, it is unknown whether Suetonius’ Chrestus is the same person as Jesus. Nevertheless, the New Testament gospels are the richest first-century sources of information on the teachings of Jesus. Much of the information regarding conflicts between Jewish Christians and other Christians in the first-century AD also comes from allusions in the New Testament; as such, my answer will heavily rely on these New Testament allusions. My answer will also be focusing exclusively on early Christian views of Jewish customs, and not include non-Christian Jewish reactions to Christianity, as your question relates to the former.
As with many of the teachings of Jesus and any religious scripture, different early Christian groups interpreted Jesus’ teachings differently. Several of Jesus’ teachings have been regarded by Jewish Christians (Christians who retained a Jewish identity) as portraying Jesus as the Jewish Messiah exclusively, with God’s covenant to Abraham remaining a covenant enjoyed only by Israel. Though God’s omnipotence and influence on politics outside Israel and Judah begins to appear in later texts of the Hebrew Bible (such as in Isaiah 10 when it is revealed God controls other nations (in this case, Assyria) for the purpose of punishing Israel), the Old Testament God is still very much the God of Israel. Even in Daniel’s account of the resurrection, Daniel is only told “your people” will be saved following the destruction of the Seleucids (Daniel 12:1). Jewish Christians believed Jesus’ arrival did not make the Torah obsolete; such sentiments could be justified in Jesus’ teachings such as his fulfillment of the law and prophets (Matthew 5:17-19). Such a teaching, especially in Matthew 5:19 “whoever breaks one of the least of these commandments, and teaches others to do the same, will be called least in the kingdom of heaven” (NRSV) can and has been interpreted as a command for Jews to continue following the laws of the Torah. There certainly was a schism between Christians who believed gentile converts should follow all of the laws of the Torah and those who believed it was unnecessary by the time of the Council of Jerusalem in the mid first-century. The central focus of the Council (as recorded in Acts 15:1-21, the only explicit primary source of the Council) was the matter of whether or not gentile Christian converts should be circumcised. Though the Council ruled in favour of those who thought physical circumcision unnecessary, the rationale for doing so does not appear to be based on Jesus’ teachings. In the account given in Acts of the Apostles, James finally convinces the Council to not rule gentile circumcision as mandatory by interpreting Amos 9:11-12 in the Hebrew Bible as prophesying the eventual belief in God by gentiles. Nevertheless, what episodes such as the Council of Jerusalem reveal is a theological divide amongst the early Christians.
Much in the same way Jesus’ teachings have been interpreted as being exclusively Jewish, they have also been interpreted as conveying the opposite, especially after the destruction of the Second Temple. The notion that God abandoned the Jews in favour of gentiles became quite prominent following the Jewish-Roman wars. The notion of invasion and destruction as divine punishment is not unique to Christians, the sins of Israel and Judah are blamed many times throughout the prophetic books in the Hebrew Bible for causing the Babylonian captivity in the sixth-century BC. Even Josephus portrays the Roman destruction of the Second Temple as retribution for Jewish tolerance of the Zealots: an extremist militaristic Jewish sect founded by Judas of Galilee in the first century AD (Antiquities, XVIII.1.1). Perhaps no New Testament verse could be better interpreted as Jesus implying gentiles are successors to the Christian faith as Matthew 21:43, which appears just after Jesus’ Parable of the Wicked Husbandmen. Jesus’ teaching “the kingdom of God will be taken away from you and given to a people that produces the fruits of the kingdom” (Matthew 21:43, NRSV) has commonly been interpreted by non-Jewish Christians as God’s abandonment of Jews for their wickedness, and gentiles as successors to Christianity (unsurprisingly, as acknowledged by David L. Turner, this interpretation has been used to justify antisemitism).
The problem with using Jesus’ teachings in the New Testament exclusively as a source of reference for official doctrine is the open-ended, vague, and unclear nature of these teachings. Just as I have given you the two aforementioned verses in Matthew (Matthew 5:17-19 and Matthew 21:43) and provided one interpretation for each verse, those same verses can be interpreted differently so that Jesus portrays the opposite meaning. Bradley Trout has argued Jesus’ teachings of him fulfilling rather than abolishing the Torah should not be interpreted as a command to continue following the Torah literally. Rather, Trout asserts, Jesus’ use of “fulfill” merely implies the prophecies in the Hebrew Bible came true during the life of Jesus, and that the Torah may be looked at in a new light with Jesus’ commands. Trout’s sentiment of not having to literally follow the Torah is shared by early Christian writers such as Paul, who is famous for his concern of converting gentiles and accommodating their needs and abilities. Alongside supporting in the Council of Jerusalem a gentile Christian’s right not to physically circumcise themselves, Paul has argued in his letters that a literal legalist interpretation of the Torah is not only unnecessary, but spiritually harmful. The Torah, Paul believes, was necessary for the Jews at a time before the coming of Jesus, though it alerted the Israelites as to what is and is not sinful, it did not prevent them from committing sin. The Torah is no longer necessary to prevent sin following the arrival of Jesus as Jesus will save those who believe in him (Romans 3:1-9, Galatians 3:19-29). Again, Matthew 21:43 has been interpreted as a new covenant between God and gentiles, but it has also been interpreted as judgment not on Jews as a collective, but on specific Jewish leaders at the time such as the Pharisees and Sadducees. This interpretation, believed by David L. Turner, is based on contextualizing Matthew 21:43 with Jesus’ Parable of the Wicked Husbandmen in Matthew 21:33-46. The parable is of a man who buys a vineyard and leased it to certain tenants. The tenants killed every debt collector the owner sent to them, including the owner’s son. According to Turner, the vineyard may be interpreted as Israel, contrasting another common Christian interpretation in which the tenants represent Israel. Using this interpretation, the tenants represent not Israel, but contemporary Israelite leaders; using this interpretation, Turner reads Matthew 21:43 as Jesus judging specific Jewish leaders in his day, and that Israel will be “given to” new righteous Jewish leaders.
It is difficult to determine for sure what Jesus’ views on converting non-Jews or following the Torah were. While it is possible Jesus saw himself as simply a Jewish reformer rather than someone disregarding the laws of the Torah, the unclear wording of many of his teachings have been used to present Christianity as a faith to be only followed by Jews, and as a faith open to gentiles, or even placing gentiles above Jews (though I believe it is highly unlikely Jesus would have viewed gentiles as superior to Jews). Many of the clearer Christian breakaways from Jewish practices come from later Christian writers, with Paul especially having a greater influence on the breakaway between Christian and Jewish practices than Jesus.
Let me know if there is anything I was unclear on or should elaborate on!
Primary Sources
The Bible (I used the New Revised Standard Version specifically as a reference)
Josephus, Antiquities of the Jews. Book XVIII, chapter 1
Suetonius, The Twelve Caesars. Divus Claudius
Tacitus, Annals. Book XV, chapter 44
Secondary Sources
Trout, Bradley. “Matthew 5:17 and Matthew’s Community.” HTS Teologiese Studies 72(3) (2016)
Turner, David. “Matthew 21:43 and the Future of Israel.” Bibliotheca Sacra 159 (2002) 46-61