Straight away both major players had a problem: Neither Rome nor Carthage had a base of operations on Sicily from which to receive reinforcements and communications from the mainland. For this reason the war would mostly be fought at sea and any land battles would be waged with the end goal of improving naval domination on Sicily. For Rome, the problem was increased by the fact that they were fighting against a superior navy.
With the expulsion of Hanno from Messana Appius Claudius Caudex was allowed to land two legions at Messana which was under siege from the combined forces of Carthage and Syracuse. The Romans broke the siege and defeated the two armies before marching south and besieging Syracuse.
The siege was brief and, with no help from the Carthaginians seeming to arrive, Syracuse sued for peace. According to the terms of the treaty Syracuse would become an ally and help supply the Roman army in Sicily. They also had to pay a light indemnity of 100 talents. This solved Rome's provisions problem. Shortly afterwards several other Carthaginian allies switched allegiance.
With a superior navy Carthage was able to set up a base of operations in Africa where they could ship their mercenary army to Sicily. The army, composed of Ligurians, Celts, and Iberians and numbered 50,000 infantry, 6,000 cavalry, and 60 elephants.
Agrigentum was a city rich from the trade of sulphur and potash. It was also strategically important and would have been used as a base of operations from which Carthage could easily move its troops along a main route along the southern coast of Sicily as well as roads north and east which led to other cities. For this reason Rome attacked Agrigentum in 262 BC.
The Roman consuls for this year were Lucius Postumius Megellus and Quintus Mamilius Vitulus. Together, with a combined force of 40,000 men, they marched towards Agrigentum.
The Carthaginian commander, Hannibal, son of Gisgo, combined his mercenaries with the local population to increase his number to 50,000. The Romans offered themselves to battle, but Hannibal refused to leave the city walls thus forcing the Romans to set up camp and begin harvesting local crops.
It was during one of these foraging missions that Hannibal made the first move. He sent a force out to harass the foragers and to disrupt the Roman's supply. The outnumbered and unarmed Roman soldiers fled to camp. It was the camp guards and watchmen who successfully repelled the Carthaginian assault.
Hannibal realised that he could not afford to lose any more men on such missions again and increasingly became reluctant to attack. The Romans, for their part, realised that they had underestimated the enemy.
One of Agrigentum's strengths as a strategically placed city was it's natural barriers: the river Hypsas to the west, the river Akragas to the east. It also sat atop a plateau surrounded by steep slopes on all sides except the west. This was also it's weakness. The Romans blockaded the city from the outside world with the intention of starving the population and forcing their surrender.
The consuls divided their armies and stationed one force near the temple of Asklepios to the south, and the other force was stationed on the road leading to and from the city in the west. Agrigentum held out for five months before supplies began to run out. Hannibal managed to get a message to Carthage appealing for help. In response Carthage sent Hanno, possibly the son of Hannibal, with reinforcements consisting, according to Polybius, of elephants, Numidian cavalry and mercenaries.
Hanno landed his troops on the south of the island at a city called Heraclea Minoa and quickly captured Herbesos, a Roman supply base. The Roman troops quickly ran short of supplies and disease soon followed. The capture of Herbesos also severed Rome's communication lines. Hanno, with the use of his Numidian cavalry, managed to trick the Roman cavalry into pursuing them and led them directly into the Carthaginian column. The losses for Rome were many. Hanno set up camp on a hill about a mile away from the Roman camp where skirmishes continued between the two armies for two months.
Morale and health must have been low in the Roman camp. With no supplies coming in from Syracuse they were at risk of starvation. The consuls had no choice but to offer battle.
Hanno refused. He intended to defeat the Romans by hunger. They would not have as much supplies as Agrigentum, yet the city had been under siege for seven months at this point and Hannibal was getting desperate. Communicating with smoke signals, Hannibal urged Hanno to relieve them from their desperate situation. Hanno had no choice but to accept battle.
The Romans defeated the Carthaginians and forced Hanno to retreat. Hannibal escaped Agrigentum by night, his rear guard harassed by Roman cavalry. Rome plundered the city and sold 25,000 (more-than-likely most, if not all, of the population) inhabitants into slavery. Though his was the Roman way at the time it proved counterproductive. Nearby cities who were wavering in their allegiances hardened their attitude towards Rome.
With this battle Rome controlled most of Sicily and secured the grain harvest for her own use. Despite the success the two consuls were denied a triumph because they allowed Hannibal to escape.