Lesson 1: When You Get Two Jews Together You Get Three Opinions.
Christians teach that Jews did not accept Jesus, and Jews claim that Jesus (and Christianity) are gentile. Both are wrong.
Judaism in the Second Temple period (when Jesus was crucified) was divided into many antagonistic factions. The main camps were the Pharisees, Sadducees, Zealots, Essenes, and Nazarenes (Christians). These groups did not get along, though they were all considered part of Judaism. Only two Jewish sects still exist (From Second Temple Period Judaism): Christians and Pharisees (Orthodox Jews). The others have (pretty much) passed away.
The Sadducees and temple priest hated the Christians (in particular). Some of these guys committed murders to obtain the position of Temple priest. Jesus called them out for their immorality. The Acts of the Apostles talk about how the Temple Priests tried to get back at Christians by trying to stop Christians from spreading the Good News that the Messiah had come.
Christianity was wildly popular with many Jews in the 1-4 centuries, but the Jewish leaders continued to hate the Christians. At the same time, the merging Roman Catholic church viewed Judaism as a rival. A Roman Catholic bishop visiting the Middle East at the end of the Fourth Century found Jews and Christians worshipping together in synagogues. Soon after the visit, the Catholic Church issued a statement forbidding Christians to worship with Jews.
In the first Century, Christians were the new kids on the block and could be kicked around by Jewish leaders. Now, the tables have turned. Establishment Judaism will have a lot more trouble keeping Christians apart from their Jewish heritage.