We don't actually know the absolute specifics for Carthage, because there are very few real literary sources from the Carthaginian perspective (I believe only one, being Hannos navigation, but I could be forgetting a few). On the other hand, we do know a great deal about how the nearby Greek kingdom of Ptolemaic Egypt acquired its elephants, and I assume that your interest is more about elephants in general than Carthaginian elephants in particular (also they were probably acquired in similar ways).

Elephants first exploded onto the Greek military imagination with Alexander's campaigns in India (technically Pakistan) where he most famously faced them in the battle of Hydaspes. After the death of Alexander, one of his successors, Seleucus, campaigned in India briefly against a bright young king named Chandragupta Maurya, also known as one of the most legendary figures in Indian history. We don't have much in the way of solid details, but the end result was that Seleucus abandoned his positions in the Greek far east and received an alliance and five hundred elephants. These elephants then played a decisive role in the Battle of Ipsus, one of the greatest battles of the Hellenistic era and a resounding success for Seleucus. Their performance in this battle seemed to touch of a mania for elephants, as the highly dynamic militaries of the Hellenistic kingdoms viewed them as the key to enduring military victory--until, of course, they were distracted by the next shiny object (to give a sense of Hellenistic military science, this is era that gave us the tessarakonteers). To skip to the end of the story, after about a century the downsides of elephants became too glaring and they faded from use (they were antiquated by Hannibal's time), and eventually all the Hellenistic kingdoms were conquered by the Romans and their boring footsoldiers.

A lot of administrative documents survive from Ptolemaic Egypt, so we can actually trace how they acquired elephants. The preference was for Indian varieties. There may be some biological reasons for that which I am not really aware of, but the primary cause would be because the elephants had a long tradition of use in warfare in India, and therefore the Indians would have been the most expert captors, tamers and trainers in the world. Acquiring an elephant from India means you are acquiring one that is ready and fit for battle, and probably comes with a mahout--elephant trainer--as well. Unfortunately politics often got in the way of this. Ptolemaic Egypt was blocked from India by the Seleucid kingdom, which means that acquiring Indian elephants could only be done at their agreement. Given that the two were frequently military rivals, this was obviously not always given, and so the Ptolemies looked to their own hinterland, that is in the southern reaches of the Nile where they still exist.

The exact day to day specifics of how this worked are unclear, but there are preserved some documents that refer specifically to elephant hunters. Whether these hunters were literally people from Egypt who went south, hunted elephants, and came back with them, or were people who went south and acquired them from local hunters is debatable, but one way or another there was a sort of administrative mercantile apparatus that brought elephants into Egypt. And we do actually have a good idea of transport: the elephants after captured were led to one of the Ptolemaic "trade centers" such as Ptolemais Theron that were founded to support their acquisition. There they were loaded onto "elephantegoi" (ships specially designed to carry elephants) and shipped north. They were unloaded at the port of Berenice, which would later because the focal point of trade between India and Rome, and taken across the desert to the Nile, where they could then be shipped up.

This process was difficult and time consuming, and so other methods were tried, including a sea route to India and a captive breeding program. The first didn't really take off until the Roman period, when traders sought pepper rather than elephants, and the latter failed for some reason that someone more familiar than me with elephant biology and behavior should talk about.

How much this applies to Carthage is uncertain, but I suspect the basics were the same, including the political difficulties of acquiring Indian elephants and the logistical difficulties of acquiring African ones. There is some suggestion that the Carthaginians were succesful in captive breeding, though. Either way, the wars against Rome pretty decisively showed they were more trouble than their worth.

A great source on the Ptolemaic trade is Sidebotham's Berenike, about the port. But it is honestly a topic with a bizarrely large scholarly literature, I guess people just love elephants.