The obliteration of Carthage by the Roman Republic at the conclusion of the Third Punic War in 146BC has done little to aid our ability to reconstruct anything about its cultural life, in the sense that we are always viewing them through the lens of the Greeks and Romans. As far as we can tell, the Carthaginians had their own literate community that wrote works about various topics, but we aren't given as much information as we'd like. As Richard Miles puts it
"...it is difficult to gauge whether the city was really a great literary centre like Athens or Alexandria" [Miles, R. 2010: 13]

Probably the most famous/widespread Carthaginian work was the 28/32-volume agricultural treatise of Mago, which was first translated into Latin but has not survived to the present, despite being cited by later Medieval Byzantine and Arabic scholars [Pliny, Natural History, 18.5; Columella, De Re Rustica, 1.10, 1.13-14]. Whether they had codified or written philosophy is pretty unclear. We do know of Carthaginians becoming relatively prominent members in Greek philosophical schools: Clitomachus, born Hasdrubal, became the head of the Academy of Plato in Athens in the mid 120s BC [Diogenes Laertius, Lives of the Eminent Philosophers, 4.67].

As was to be expected of a famously seafaring people, it is unsurprising to find out that we have records of sea voyages, referred to as the periplus. Pliny the Elder refers to at least two different authors: Himilco, who wrote his account of his journey along the English Channel and Northern Gaul, and Hanno, who sailed from Spain to Arabia (Pliny the Elder, Natural History, 2.169). The claim of Hanno is more controversial, because there is an apparent Greek translation of a Punic inscription from Hanno which recounts how he might have explored the West African coastline as far as modern Gabon [for a translation of the inscription, see Livius]

In terms of histories a la a Carthaginian equivalent to Herodotus or Livy, there is some evidence, though nothing survives. In Numidia (a neighboring North African kingdom to Carthage) their more literary-minded rulers like Hiempsal and especially Juba II used Carthaginian histories when writing their own accounts of Numidia and the peoples of the surrounding region [Pliny, Natural History, 18.5; Sallust, The Jugurthine War, 17.7]. One of the most cited authors of the First Punic War was Philinus of Agrigentum/Akragas, who composed a history on the Sicilian conflict that was very pro-Carthaginian, something that was heavily criticized by the (mildly pro-Roman) historian Polybius of Megalopolis [Polybius, The Histories, 1.14-15]. Philinius was a Sicilian-born Greek, but the slant with which he writes may indicate that he had access to Carthaginian writings contemporary to the period, though we aren't told by Polybius.

We know that Carthage had libraries, or at least the approximate equivalent of them. The only book of interest to the Romans was that of Mago's Agricultural Treatise, which was specifically preserved and translated on Senatorial decree [Columella, De Re Rustica, 1.13-14]. The fates of the rest is attested to by Roman authors: according to those like Pliny and Sallust, much of their contents were distributed to neighboring North African kingdoms, more specifically the Numidians [Pliny, Natural History, 18.5; Sallust, The Jugurthine War, 17.7]. While we don't have the writings of the Numidian rulers either, they were also cited by later authors who thus indirectly perpetuated the Punic viewpoints of the earliest histories of North Africa. The only Carthaginian work of interest to the Romans was Mago's Agricultural treatise, which was sought out above all else during the sack of the city. In the 4th century AD, St. Augustine of Hippo (who was himself of Punic origin and could speak the language) wrote a defense of the Punic language and directly referred to the existence of books written in Punic (St. Augustine, Letters, 17). While it is unclear that he was referring to works written by the Carthaginians or from authors closer to his own time, it is a testament how the Punic literary traditions continued to survive well after its incorporation into the Roman Empire.

St. Augustine - Letters
Columella - De Re Rustica
Diogenes Laertius - Lives of the Eminent Philosophers
Pliny - Natural History
Polybius - Histories
Sallust - The Jugurthine War

Hoyos, D. 2010 The Carthaginians
Lendering, J. Hanno the Navigator, from Livius.org (Link)
Miles, R. 2010 Carthage Must Be Destroyed: The Rise and Fall of a Civilization