The Big Split - Opposition from Jewish Leaders
From the beginning of Jesus' ministry there was animosity from some of the leaders in the Jewish community. Although Jesus and the early apostles had widespread support from the man-on-the-street, there were certain elements who opposed them.
Some of the Pharisees, though by no means all, were upset at Jesus preaching against hypocrisy in their ranks, even though affirming their teaching (Matt. 23:3). There continued to be good relations between Jesus and many of the Pharisees to the extent of them inviting him for meals (Luke 14:1) and warning him of Herod's plots (Luke 13:31). And in the early chapters of Acts it is the Pharisees who supported the right of Peter (Acts 5:34) and later Paul (Acts 23:9) to teach the people. Paul remained a Pharisee all his life, and Jesus' teaching had much in common with the Pharisees' especially the school of Hillel.
The real opposition was among the Saducees and the temple leadership. There is a noticeable lack of mention of the Pharisees during the events of Jesus' final days. But it was the Saducees and the priests who engineered the plot to arrest and try him illegally. Remember that these priests were not the hereditary priests from the line of Aaron. The high priesthood at this time was an appointed position, and frequently the person with the most money got appointed by Herod and others. There was corruption rather than spirituality among the priests and temple officials and Jesus had spoken out against it repeatedly. So it was no great surprise that while the populace was preoccupied with observing Passover, a relatively small group of priests, Saducees, and temple officials arrested an tried Jesus.
These same people were the ones who opposed the early apostles as well. The early believers had continued to worship in the temple for the Jewish communal prayer times (Acts 2:46; 3:1). Soon the temple officials got fed up with their teaching and had them arrested (Acts 4:1; 5:17). But they were persuaded by Gamaliel to let them go.
The opposition to Stephen seems to have come from a particular synagogue of Jews who had moved to Jerusalem from the Diaspora. When they couldn't successfully oppose him in public debate, they persuaded false witnesses to say that he opposed the temple and the Torah, things that Stephen never claimed. These scheming manipulators managed to convince many, including Saul, to oppose the apostles teaching. It was unusual for Saul, as a Pharisee, to persecute the believers, but he gained legitimacy by appealing to the Saducean priesthood (Acts 9:1).
The Big Split - Inclusion of Gentiles