1. Answer(s): Paul of Tarsus and other disciples; Constantine’s conversion; outlawing of public non-Christian sacrifices by Theodosius (d. 395)

2. Converting the Gentiles
Paul believed God sent him to convert non-Jews, or Gentiles
Paul helped make Christianity broader religion, attracted many new followers
Helped establish Christian churches throughout eastern Mediterranean
Paul’s epistles, or letters, to those churches later became part of the New Testament
Paul found some Jewish customs hindered missionary work among non-Jews, dispensed with those requirements for Christians
Paul emphasized new doctrines that helped distinguish Christianity from Judaism
Christianity spread; message of love, eternal life after death found appealing
By AD 300, some 10 percent of Roman people were Christian

3. What is the difference between Christianity and Judaism?

4. Of the major world religions, Christianity and Judaism are likely the most similar. Christianity and Judaism both believe in one God who is almighty, omniscient, omnipresent, eternal, and infinite. Both religions believe in a God who is holy, righteous, and just, while at the same time loving, forgiving, and merciful. Christianity and Judaism share the Hebrew Scriptures (the Old Testament) as the authoritative Word of God, although Christianity includes the New Testament as well. Both Christianity and Judaism believe in the existence of heaven, the eternal dwelling place of the righteous, and hell, the eternal dwelling place of the wicked (although not all Christians and not all Jews believe in the eternality of hell). Christianity and Judaism have basically the same ethical code, commonly known today as Judeo-Christian. Both Judaism and Christianity teach that God has a special plan for the nation of Israel and the Jewish people Read more:

5. The all-important difference between Christianity and Judaism is the Person of Jesus Christ. Christianity teaches that Jesus Christ is the fulfillment of the Old Testament prophecies of a coming Messiah / Savior (Isaiah 7:14; 9:6-7; Micah 5:2). Judaism often recognizes Jesus as a good teacher, and perhaps even a prophet of God. Judaism does not believe that Jesus was the Messiah. Taking it a step further, Christianity teaches that Jesus was God in the flesh (Hebrews 1:8). Christianity teaches that God became a human being in the Person of Jesus Christ so He could lay down His life to pay the price for our sins (Romans 5:8; 2 Corinthians 5:21). Judaism strongly denies that Jesus was God or that such a sacrifice was necessary.

6. Jesus Christ is the all-important distinction between Christianity and Judaism. The Person and work of Jesus Christ is the one primary issue that Christianity and Judaism cannot agree upon. In Matthew 15:24, Jesus declared, "I was sent only to the lost sheep of Israel.” The religious leaders of Israel in Jesus' time asked Him, “Are you the Christ, the Son of the Blessed One?” Jesus replied, 'I am,' ... And you will see the Son of Man sitting at the right hand of the Mighty One and coming on the clouds of heaven" (Mark 14:61-62). But they didn’t believe His words or accept Him as the Messiah.

7. DOCTRINES OF TRINITY

8. Trinity, in Christian doctrine, the unity of Father, Son, and Holy Spirit as three persons in one Godhead.
Neither the word Trinity nor the explicit doctrine appears in the New Testament, nor did Jesus and his followers intend to contradict the Shema in the Hebrew Scriptures: “Hear, O Israel: The Lord our God is one Lord” (Deuteronomy 6:4). The earliest Christians, however, had to cope with the implications of the coming of Jesus Christ and of the presumed presence and power of God among them—i.e., the Holy Spirit, whose coming was connected with the celebration of the Pentecost. The Father, Son, and Holy Spirit were associated in such New Testament passages as the Great Commission: “Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit” (Matthew 28:19); and in the apostolic benediction: “The grace of the Lord Jesus Christ and the love of God and the fellowship of the Holy Spirit be with you all” (2 Corinthians 13:14). Thus, the New Testament established the basis for the doctrine of the Trinity.

9. The doctrine developed gradually over several centuries and through many controversies. Initially, both the requirements of monotheism inherited from the Hebrew Scriptures and the implications of the need to interpret the biblical teaching to Greco-Roman religions seemed to demand that the divine in Christ as the Word, or Logos, be interpreted as subordinate to the Supreme Being. An alternative solution was to interpret Father, Son, and Holy Spirit as three modes of the self-disclosure of the one God but not as distinct within the being of God itself. The first tendency recognized the distinctness among the three, but at the cost of their equality and hence of their unity (subordinationism); the second came to terms with their unity, but at the cost of their distinctness as “persons” (modalism). It was not until the 4th century that the distinctness of the three and their unity were brought together in a single orthodox doctrine of one essence and three persons.

10. The Council of Nicaea in 325 stated the crucial formula for that doctrine in its confession that the Son is “of the same substance as the Father,” even though it said very little about the Holy Spirit. Over the next half century, Athanasius defended and refined the Nicene formula, and, by the end of the 4th century, under the leadership of Basil of Caesarea, Gregory of Nyssa, and Gregory of Nazianzus (the Cappadocian Fathers), the doctrine of the Trinity took substantially the form it has maintained ever since. It is accepted in all of the historic confessions of Christianity, even though the impact of the Enlightenment decreased its importance.

11. In 325 AD, Constantine called the Council of Nicea so that the bishops could work out their differences. They declared the Son (Christ) to be of "one substance" with the Father.
The Nicene Creed
We believe in one God, the Father, the Almighty, maker of heaven and earth, of all that is, seen and unseen.  We believe in one Lord, Jesus Christ, the only Son of God, eternally begotten of the Father, God from God, light from light, true God from true God, begotten, not made, of one Being with the Father; through him all things were made. For us and for our salvation he came down from heaven, was incarnate of the Holy Spirit and the Virgin Mary and became truly human.  
For our sake he was crucified under Pontius Pilate; he suffered death and was buried. On the third day he rose again in accordance with the Scriptures; he ascended into heaven and is seated at the right hand of the Father. He will come again in glory to judge the living and the dead, and his kingdom will have no end. 
We believe in the Holy Spirit, the Lord, the giver of life, who proceeds from the Father [and the Son], who with the Father and the Son is worshiped and glorified, who has spoken through the prophets. We believe in one holy catholic and apostolic Church. We acknowledge one baptism for the forgiveness of sins. We look for the resurrection of the dead, and the life of the world to come. Amen.

12. Scriptures? Hebrew Scriptures “Protestant” OT: 39 books
The Tanakh (TNK) is an acronym of the initial Hebrew letters of the Tanakh's three traditional subdivisions: Torah (Law) Nevi'im (Prophets) and Ketuvim (Writings). According to Jewish tradition, the Tanakh consists of twenty-four books.
“Protestant” OT: 39 books
“Catholic” OT: 46 books add 1,2 Maccabees; Wisdom, Ecclesiasticus; Baruch (Deutero-canonicals)

13. Old Testament, New Testament
Jewish Scholars have no issue with this nomenclature.
The Christian Old Testament (39, 46 books) is not the same as the Hebrew Scriptures (24 books)
The structure of the Jewish Sacred Writings is not the same as that of the Christian Old Testament.
The Jewish Texts are structured so as to point to reconstruction and renewal of Israel.
While all Christians agree there are 27 books in the New Testament, Catholics (46)and Protestants (39) disagree on the number of books of the Old Testament.
Protestant scholars follow Martin Luther who accepted only those texts originally written in Hebrew, rejecting the texts first recorded in Greek. Hence seven Deutero- (extra)canonical books of the Catholic Bible (Septuagint)
See Amy –Jill Levine on Jews and Christians and the Old Testament in The Misunderstood Jew: The Church and the Scandal of the Jewish Jesus, 2007.