Why did the Lord show such a difference between the Syro-Phoenecian woman and the Roman Centurion, when they were both Gentiles that served other gods?

You are assuming too much about whom both parties worshipped. The Gentiles worshipped Jehovah through the patriarchal system and the Jews worshipped God according to the Law of Moses which was given only to the Jews. However, that is of little consequence as far as the lesson goes. Jesus was drawing the woman out as to her faith in Him for the benefit of those who were accompanying Him. The Centurion had already demonstrated his faith, thus he is commended and his servant is healed. The big lesson here was that these Gentiles believed on Jesus more that the Jews through whom he had come. Even though Hebrew prophets spoke accurately about Jesus the Jews were reluctant to accept Him as the Messiah because He did not fit their preconceived ideas concerning Him. These two Gentiles showed they were more receptive of the Messiah (a Jew) than were the Jews.

What is a Nazarite?

A Nazarite was a person who, under ancient Hebrew law bound himself or herself to abstain from wine and all products of the grape; to wear the hair long and uncut, and to observe purification in abstaining from contact with the dead.  If they witnessed death or otherwise approached death, their consecration had to be entirely recommenced.  The period of observance of this vow varied from eight days to a month, or even a life time (e.g. John the Baptist).  When this period had concluded, offerings were made, the hair was cut off and burnt, and the Nazarite was discharged from his vow (Numbers 6; Amos 2:11,12)

Were the Gentiles called Christians and the Israelites called Jews?

 The designations Jew, who were Israelites because they were descendants of Israel, and Gentile when used in the Bible were terms used by the Jews to distinguish themselves from everyone else in the world.  That is how the Jews viewed things; if you were not a Jew then you were a Gentile.  Another expression that is sometimes found in the New Testament is a reference to the Jews and Greeks, such as in Romans 1:16.  This designation fits the same purpose of distinguishing Jews from all others.  These designations would also help the Gentiles understand the glory of their having been made a part of God's people.  Note especially Ephesians 2:11-22.  Where God's covenants and promises had previously been given to the Jews only, through Christ the Gentiles would also be participants in them.  However, these designations, Jew and Gentile or Jew and Greek, would not be something that would continue, for in Ephesians 2:15 the Bible says that God made the Jews and Gentiles into one new man.  He didn't make them all Jews and He didn't make them all Gentiles.  He made them Christians.  To summarize then, Gentiles were Gentiles, Jews were Jews or Israelites, and converts from either group were called Christians.

Please explain to me why the Jews did not care for Jesus. Also, do they still feel that way today?  

We might not want to say that the Jews as a whole rejected Jesus.  It is true that He was rejected by many, and maybe even most Jews, but not all.  There were those who accepted His teachings and became His followers.  Acts 1:15 states that there were 120 disciples who were gathered together when Matthias was selected to replace Judas.  Also, 1 Corinthians 15:6 says that Jesus appeared to over 500 brethren on one occasion after His resurrection. 

It was mainly the scribes and the Pharisees who rejected Jesus because He challenged the fact that they were following their own traditions rather than what God had truly meant in the Law.  See Matthew 23 for one such discourse.  There was another problem also.  The Jews were looking for a Messiah to restore the literal throne of David, Acts 1:6, which, as this text plainly indicates, included His own disciples.  However, Jesus did not come to restore or set up an earthly kingdom, John 18:36.  Because Jesus did not fit their picture of the Messiah and because He called for an inner holiness, and not just an outward show, He was rejected.  The Jews view Jesus today more or less as a great teacher of that day, or as a prophet, but certainly not the Messiah.  There is not an antagonism toward Him today as there would have been then.

Where in the Bible does it discuss Christ's salvation for the Jew and Gentile?
Where does it explain that the Jews are now under the new   law of Jesus Christ?

When Joseph and Mary brought the baby Jesus to the Temple. the Holy Sprit said through Simeon, "For mine eyes have seen thy salvation, which thou hast prepared before the face of all peoples; A light for revelation to the Gentiles, And the glory of thy people Israel" (Luke 2:30-32).  Prophecies had revealed that Gentiles would participate in the same blessings as the Israelites through Christ. (Isa. 42:6; 49:6.)

The gospel was preached first to the Jews and then to the Gentiles. The great and final commission of Christ was that the gospel was to be preached to every person (Mark 16:15, 16.) This they did. "And when they were come, and had gathered the church together, they rehearsed all that God had done with them, and how he had opened the door of faith unto the Gentiles"
(Acts 14:27). "That the Gentiles should be fellow-heirs, and of the same body, and partakers of his promise in Christ by the gospel" (Eph. 3:6-8).

What is the time of the Gentiles. Did it have a beginning and end time? I heard a man on the radio and he said it began in Daniel's time?

One of the first, if not the first mention of the "Gentiles" is in Gen. 10:5 regarding the descendents of Noah, and the last in Rev. 11:2. It was term used by Jewish people to refer to foreigners, or any other people who were not Jewish. The Jews developed a feeling of exclusivism over a period of several centuries.

In early Hebrew history, Gentiles or non-Jews were treaded cordially by the Israelites (Deut. 10:19; Num. 35:15; Ezek.. 47:2) Men of Israel sometimes married Gentile women, such as Ruth and Bathsheba. However, after the Hebrews returned from Babylon captivity, the practice of intermarriage was discouraged (Ezra 9:12;10:2-44; Neh. 10:30). Separation between Jews and Gentiles became increasingly strict; by the New Testament period the hostility was complete. Jesus said the wall between Jew and Gentile was to be broken down. "There is neither Jew nor Greek...for you are all one in Christ Jesus". Gal. 3:28.

What is the difference between a Jew and a Christian?

A Jew is one who considers Jesus Christ as no more than a later prophet or teacher. The Jew still looks for a messiah to come. A Christian accepts Jesus as the Christ, the Son of God, Matthew 1:18-23; John 1:1-12; 2 Corinthians 5:19; Colossians 1:13-17; Hebrews 1:1-14; and 1 John 5:1-13. 

The Jew follows only what the Christian knows as the Old Testament. The Christian lives under the Law of Christ, Romans 7:1-8:4; Galatians 2:16; 3:10-14; Colossians 2:14; Hebrews 7:18-22; 10:1-10.