What was Rome's objective in fighting Carthage during the Punic Wars?

Answer 1

It started out as a dispute over influence in Sicily, then turned into full-scale conflict. As wars between great powers are wont to do, this conflict grew way out of proportion to the original flashpoint, and soon the Romans sought to end the Carthaginian state as a rival great power.

A peace treaty similar to the 1919 Peace Treaty ending the Great War was ratified, and soon there were those in the defeated city-state that wanted a revision of the terms: to throw out the treaty and revive Carthage as a power. Hannibal's subsuquent war with Rome convinced the Romans that Carthage really had to be ended, and that was meant to be taken literally. After final victory over Hannibal, Rome sought to make Carthage politically and economically dependent, but some Romans thought that was still too little. Cato the Elder started ending each speech in the Senate with "Carthage must be destroyed!" (at one point, he dropped a huge fig leaf onto the floor. When other senators gaped in awe at its size, Cato reminded them that the monster leaf came from a state only three days away by sea.) Soon Rome provoked a third and final war with her bitter enemy, and made good on its intention to crush Carthage for good.

Answer 2

Business.
Carthage had very good trade connections and outposts all around the med. The Carthaginians had settlements from Lybia to Spain and a efficient fleet. Of course the Romans wanted their power.

Answer 3

It started as a trade dispute and became a vendetta.