Unfortunately, there is no surviving Punic literary tradition, so we typically only know what Carthage is doing if they're doing it to the Greeks or the Romans. For the duration of the Classical period (490-323 BC), Rome was at best a regional power contained in Latium, so the relevant events are to do with Carthaginians interacting with the Greeks.

It just so happens that they interacted with Greeks a lot, and specifically around the times you're asking about. Phoenicians and Carthaginians first established settlements on the north and west of Sicily in the early 8th century BC, soon followed by the first Greeks settling the east coast. In the century that followed, the Greeks founded cities all along the east, north and south coasts of the island, with Carthaginian assuming control over much of the west. The original population of Elymians, Sikels and Sikanians was either enslaved or forced to retreat inland. The result was an island divided between expansionist clusters of Greek and Carthaginian city-states, each with designs on the rest of this rich and fertile landmass. Sicily was to become the great battleground at the western end of the Greek world.

In 480 BC, as Xerxes and his army marched on Greece, the alliance of defiant Greek states sent an embassy to Gelon of Syracuse, arguably the most powerful man in the Greek world. Having united the large and prosperous Sicilian cities of Gela and Syracuse under a single tyranny, Gelon commanded unrivalled resources that the defenders of mainland Greece badly needed to hold off the invaders. And what they asked, Gelon offered; he promised to provide no fewer than 20,000 hoplites, 2,000 cavalry, and thousands of light-armed troops for the cause. He had only one condition: that he be allowed to lead the coalition force. The Spartans, who took their own hegemony as given, rejected this out of hand. Gelon replied that if they would not let him lead them, they had better look for help elsewhere.

This is the story as Herodotos tells it. But he goes on to explain that it was just as well that Gelon didn't send his army abroad, because right as Xerxes entered Greece, the Carthaginian general Hamilcar attacked with an army of 300,000. The Sicilian Greeks needed every man they had to defend their homes. We may assume that when Gelon refused to help the Greeks, it was mainly because he knew that a Carthaginian campaign was imminent, and that he wouldn't be able to spare men for a losing war in Greece. In the event, and against all odds, the anti-Carthaginian alliance of Syracuse and Akragas won a spectacular victory at Himera, shattering the Carthaginian invasion force before it could do serious damage to the Greek cities on Sicily. Herodotos (7.166) claims that this battle was fought on the very same day that the Greeks defeated the Persian fleet at Salamis. However, Diodoros of Sicily (11.24.1) tells us it was fought on the same day as the battle of Thermopylai.

This Diodoros, too, is most adamant of all ancient sources that the timing of Hamilcar's attack was no coincidence. He insists (11.1.4-5) that Xerxes, once he was persuaded to conquer Greece, sent an embassy to Carthage to arrange for a joint invasion of the Greek world, from west and east simultaneously. This explained the story that the decisive battles had taken place on the same day. Modern scholars don't put much stock by this conspiracy theory, and prefer to explain the two invasions as independent ventures years in the making. But the allegation that Persians and Carthaginians were in cahoots to overrun the Greeks in 480 BC is a perfect expression of the Greek idea that they lived in a dangerous world, surrounded by hostile barbarians, with only their arms and their gods to protect them.

The century and a half between the Persian Wars and the reign of Alexander saw an endless see-saw of domination over Sicily. Carthage held the upper hand at the end of the 5th century BC, razing the great cities of Selinous, Akragas and Gela to the ground - but they lost their army at Syracuse and were forced to give up nearly all their holdings on the island to the tyrant Dionysios of Syracuse in the ensuing decades. Their fortunes revived several times, and it took the combined effort of all Greek communities on the island to ward off the Carthaginian threat - but the Carthaginians could not breakthe supremacy of Syracuse. It took until the middle of the 4th century BC for the largest Greek city to weaken, and for Carthage to get a chance to regain what it had lost. In the 330s BC, Carthage might have been an excellent ally for the Persian king Darius III in his efforts to deflect some of the unrelenting force that was Alexander III. Might - if not for a Corinthian called Timoleon.

In 340 BC, in a half-hearted response to a plea from the subjected populations of Sicily, Corinth sent the senior citizen Timoleon with a tiny force of volunteers to liberate the Greek cities of Sicily from the tyrants that ruled them. Against all expectations, Timoleon promptly did so. Then, noticing that the Carthaginians spied Greek weakness and were gathering an army, he marched out to confront this invasion force and annihilated it in battle at the river Krimesos in 339 BC. Sent into headlong rout, the Carthaginians once again lost all but a few western footholds on Sicily. It has been suggested that in 338 BC, when Philip II formed his League that bound the Greek states to support his war on Persia, he deliberately did it in the city of Corinth because of its link to Timoleon's recent decisive victory against the barbarians of the west.

The result was that at the time of Alexander's reign (336-323 BC), Carthage was in no condition to present a credible threat to the Western Greeks. We consequently hear nothing about them until the final decade of the 4th century BC, when the rise of Agathokles as the new tyrant of Syracuse prompted them to march again.