What is the purpose of the quarrel between the senators and the commoners in the beginning of Julius Caesar?

Answer 1

They are actually tribunes, not senators. They are angry with the tradesmen for illegally taking a holiday to celebrate Caesar's victory over Pompey. Marullus contends that a victory in a civil war does not deserve a triumph. He argues, "Wherefore rejoice? What conquest brings he home? What tributaries follow him to Rome to grace in captive bonds his chariot wheels?" As for the tradesmen, they don't care, so long as they get to party on. What is the purpose of all this?

1. Shakespeare's opening scenes are often intended to settle the audience down and get their attention. They frequently involve characters who will never be seen again (think of the opening of Romeo and Juliet), and rarely involve the lead actor (Richard III is an exception). Here the entire scene involves characters who are insignificant in themselves, but there is conflict and humour (from the funny cobbler) which should get the audience's attention before the really important stuff starts happening with the introduction of Caesar, Antony, Brutus and Cassius in Scene 2.
2. Caesar is shown to be one of those politicians who courts the support of the mob. He supports their having an illegal triumph for him so they can have a party, even though it disrupts the commercial life of the city. It also demonstrates his vanity. Caesar's populist approach will be emulated famously by Antony with his appeal to the mob at Caesar's funeral.
3. The scene sets up a little remark in the next scene. Casca, after saying that whatever Cicero said was Greek to him says, "I could tell you more news too: Marullus and Flavius, for pulling scarfs off Caesar's images, are put to silence." That is to say, Caesar had them murdered. Now, the undercurrent is that the two tribunes had some sympathy for Pompey, which was probably the real reason they were killed, but even so, this casually brutal behaviour marks Caesar as a bloodthirsty tyrant, foreshadowing the casual and cruel behaviour in Act IV Scene 1

Answer 2

It shows some senators dislike Caesar's popularity, using the commoners' clothing as an excuse. The tradesmen, instead of working, are roaming the streets celebrating Caesar.