What are the major conflicts of the play Julius Caesar?

There are three main conflicts in this play. 

First, there is the conflict of friendship. Brutus and Caesar were friends. It was actually more than this. Caesar was like a father to Brutus. So, for Brutus to betray Caesar, there was a huge internal conflict. This is why if there is a tragic hero in the play, it is Brutus. 

Similarly there was a conflict of national interest. Brutus was torn in his loyalties, because he strongly believed in and valued the Roman republic and he loved Caesar. In the end, he choose to defend the Republic (wrongly). There is an important historical detail to keep in mind, because when the republic was established, it was Brutus's ancestor (another Brutus) who started the consulship and the republic. So, his family line was steeped in republicanism. 

Finally, there is a conflict between religion and reason. Right from the beginning of the play there were religious omens. For example, the soothsayer repeatedly said "Beware of the ides of March." There was also the storm and the dream of Caesar's wife. Caesar did not heed any of these things.