Well first off this depends on which of the Punic Wars we are talking about, so lets look at both.

We often try and put some big emphasis on big motives like the conquest of Rome like you suggest or establishing an empire etc, etc. However many wars happen through random circumstance, luck, misunderstandings etc. This is the case with the First Punic wars. To briefly explain, a group of Italian mercenaries called the Mamertines (named after the Oscan god of war Mamers- Roman Mars) were out of employment. Eventually the city of Messana (now Messina) in north eastern Sicily offered to hire them to protect the city. After a short time the Mamertines decided they quite liked the town and promptly slaughtered the population of the city. They proceeded to rule the city for some 20 years, using it as a base of operations from which to raid the surrounding areas, becoming pirates on land and sea. This did not go unchallenged forever, in 270BC the Mamertines came into conflict with one of the major two powers in Sicily at the time Syracuse(the other being Carthage), led by the famous tyrant Hiero II. Hiero inflicted a devastating defeat on the Mamertines and in 265BC began to march on the city. The Mamertines called to Carthage for aid, and the Carthaginians swiftly moved a small garrison into the city. This was not unusual for Sicily, Carthage and Syracuse (the leader of the Greek cities) had battled for control of the island for centuries at this point, although lately the Carthaginians had been in the ascendancy. The Mamertines got fed up with the Punic garrison that had just saved them however and so asked the Romans for protection.

There was much debate in Rome about what to do. Many Romans worried about what might happen if Carthage established control of all Sicily, many thought of the great loot and victories that could be achieved in Sicily (this was particularly pushed by the years consul), others thought Messana a good base of operations for further expansion. However some Romans and certainly our source Polybius sides with them, believed this act completely dishonorable and opportunistic. The Mamertines were a bunch of criminals, who were asking for help in betraying the guys who just saved them. Furthermore the Romans had just recently fought a group of similar pirates in Rhegium to "free" the locals and show Roman fides, it would be rather hypocritical to then go and help another group of pirates just across the water.

In the end the war party won, and a Roman consular army managed to sneak across the strait and push back the small Carthaginian and Syracusan forces at the city. At this point no side seems to have had any plans greater than their limited strategic aims, and neither side thought a war would be fought until the consular army actually crossed. Both the Romans and Carthaginians thought a show of force (Roman army, Carthage navy) would overawe their opponents into backing down. Polybius states that the Romans only increased their own aims to include the conquest of all of Sicily when the achieved a victory over the Carthaginians at the city of Agrigentum.

So clearly there were no great aims for the First Punic War. Both sides sort of happened into a large scale war, with their war aims and efforts growing organically with the conflict.

The situation is not entirely dissimilar for the Second Punic War. While obviously the Carthaginians were more prepared for a large war, both sides hoped to end the war with a swift strategic strike (Carthage in Italy, Rome in Spain and Africa). The Carthaginians ended up in the conflict more due both to their grudge with Rome over the taking of Sardinia, the Roman interference in Spain, and a feeling that the Romans treated them not as equals but as a subordinate state.

In terms of Carthaginian war aims these are hard to determine as we have no direct punic sources on the war. However Hannibal's campaign, Carthage's conduct of the war, and the treaty with Philip V of Macedon do shed some light on the Issue. Carthage launched invasions to reconquer Sicily and Sardinia, so these could reasonably be considered part of the war aims. However Hannibal's Campaign clearly did not envision the destruction of Rome, his treaty with Philip V of Macedon makes clear that they expected Rome to still be around and capable of attack after the war, pledging to protect each other and their allies against further Roman aggression. While Hannibal had to be lenient with the cities that switched to his side, in order to entice them to switch to his side, its still pretty conspicuous that the treaties essentially grant these cities independence and even imply that they will gain land from the war. There is no suggestion of a continued Carthaginian presence in Italy after the war, although this is muddled, as Hannibal does supposedly promise his soldiers Italian land at one point, but this may just be the usual general stuff as he also promised them Spanish and African Land an Punic citizenship at certain points.

In conclusion Carthaginian war aims during the second war are impossible to pin down with any certainty. They almost certainly had little interest in the direct conquest or subjugation of Rome itself, and indeed it is implied in treaties that they expect Rome to continue to exist in some form as an independent state after the war. They partially simply wanted to regain honour and status lost during the First war, and avenge their defeat. They probably wanted to retake Sardinia and Sicily. They may have wanted to establish client states or allies out of the Cisalpine Gauls and the southern Italian states in order to neuter Rome. However again any certainty on these war aims is impossible, as not even our ancient sources know the war aims of the Punic state, you would have to ask Hannibal yourself.

Sources:

Adrian Goldsworthy Fall of Carthage

John F. Lazenby The Hannibalic War

Nigel Bagnall The Second Punic War

Dexter Hoyos Mastering the West Rome and Carthage at War

Polybius The Histories

Livy The history of Rome