What are the main events in Julius Caesar by Shakespeare?

The first important event in the play is the Feast of Lupercal. Cassius talks Brutus into joining a conspiracy against Caesar. Meanwhile, Mark Antony offers Caesar a crown, which he refuses three times. This irritates Casca, Cassius, Brutus, and the others.

The conspirators gather at Brutus’s house. Having talked himself into killing Caesar, Brutus takes control of the group. He overrules Cassius in several decisions, including who should die aside from Caesar. Brutus wants to ensure the conspirators' actions are seen as noble.

Let us be sacrificers, but not butchers, Caius.
We all stand up against the spirit of Caesar;
And in the spirit of men there is no blood:
O, that we then could come by Caesar's spirit,
And not dismember Caesar (Act II, Scene 1)! 

Caesar almost doesn’t make it to the senate on the Ides of March. His wife has a seemingly prophetic dream that he is in danger and urges him not to go. This supports the soothsayer’s warning that Caesar should “Beware the Ides of March.” Decius Brutus convinces Caesar the dream means something good, so Caesar agrees to go.

The conspirators gather around Caesar, pretending to have a suit for him. Casca stabs him first, followed by all the others. Brutus is the last person to stab Caesar, and Caesar asks him, “E tu, Brute?” Antony requests an audience with Brutus, and Brutus grants him safe passage. Antony shakes all the conspirators’ bloody hands, and gets permission from Brutus to speak at Caesar’s funeral after Brutus.

Brutus speaks to a welcoming crowd, telling them they are now free of the tyrannical Caesar. Antony then speaks, telling the crowd that the supposedly honorable men murdered Caesar. He shows them Caesar’s will. The angry crowd turns into a violent mob, targeting the conspirators and running them out of town.

Civil war breaks out from there. Antony, Octavius (Caesar’s heir), and Lepidus are on one side; Brutus and Cassius are on the other. Antony’s army leads a proscription to pay for its war, killing enemies or just taking their property.

At Philippi, Octavius and Antony are more successful than Brutus and Cassius. Cassius did not want to go to Philippi, and Brutus even had a visit from Caesar’s ghost about not going. Still, Brutus insisted they go there to fight Antony and Octavius. Cassius commits suicide after mistaking a victory for a defeat. Brutus then commits suicide. Antony says Brutus was the noblest Roman of them all.

What are some significant incidents in Shakespeare's Julius Caesar?

This is a good question. There are several very important incidents in the play Julius Caesar.

First, there are many religious incidents that are important. The words of the soothsayer - beware the ides of March - and Caesar's disregarding of these words are significant. This shows that Caesar might be filled with hubris. The same can be said when his wife asks him not to go to the Senate on account of a bad dream. 

Second, another key incident in the play is when Cassius persuades Brutus to join the conspiracy. To have Brutus on board makes the conspiracy more noble, because Brutus is an honorable man and one who is synonymous with the Republic. 

Third, another key incident is when Antony is allowed to give a speech during Caesar's funeral. Through his speech, especially his use of the word, "honorable," he is able to turn the Romans against the conspirators. 

Finally, the death of the conspirators and the rise of Octavian is significant, as this shows the end of those who killed Caesar, but also the rise of Octavian, who will later be called Augustus.