Israel is God's chosen people, which is largely made up of Jews, but it also includes Gentiles who have been grafted into Israel through faith in Messiah (Romans 9:6-8, Ephesians 2:12-19). However, Gentiles do not need to become Jewish proselytes in order to do so and according to Paul, Jews should remain Jews and Gentiles should remain Gentiles (1 Corinthians 7:18). At its roots, Christianity was a sect of Judaism, so it was no more distinct from Judaism than Protestants and Catholics are distinct from Christianity. After all, Christians are following a Jewish Messiah, who came in fulfillment of Jewish prophecy, and who did not come to start his own religion, but bring fullness to Judaism. For roughly the first 7-15 years after Messiah's resurrection up until the the inclusion of Gentiles in Acts 10, all Christians were Jews, so there is not necessarily a division between the two, which means that Jews did not have to give up their Jewishness in order to follow their Jewish Messiah. So people are Christians because they have faith in Christ and seek to follow his example and people are Jews because they were born Jews or have become Jewish proselytes.
I recommend listening to this intro to Galatians which details the groups of people that were around during that time, as well as the rest of it if you're interested in a good study:
Galatians- Messianic Jewish audio teachings by Stan Farr
