The Carthaginians spoke a dialect of Phoenician that we now call Punic; however, the Carthaginians referred to it as "chanani", or Canaanite- the Phoenicians were, incidentally, among the peoples known in the Hebrew Bible as Canaanites.

The Canaanite languages also include Hebrew, which was actually quite closely related to Phoenician. In turn, Canaanite was a subdivision of the Semitic languages.

We actually know what consonants Phoenician had pretty well, since they invented an abjad (the Phoenician "alphabet") that included one letter for each consonant in their language. However, we don't know what each of those consonants actually sounded like. We also don't know the vowels particularly well, since Phoenicians generally didn't bother to write them. Instead, we have to infer the vowels from transcriptions in alphabets that had vowels (like the Greek or Roman alphabet) and from etymology, since we have a pretty good understanding of what the proto-Semitic precursor of Canaanite was like. Well, a pretty good understanding for a language spoken 5000-6000 years ago!

We know that Phoenician had a /b/, /d/, and /g/ sound, as well as /p/, /t/, and /k/ sounds. In Punic, at some point, /p/ softened to an /f/ sound; /t/ and /k/ may have also softened to a /th/ and a /kh/ sound, though the transcriptional evidence is ambiguous, since neither Latin nor the Greek of the time had /th/ or /kh/.

It also had an /h/ sound and a glottal stop /ʔ/, like the dash in uh-oh. Additionally, it had two pharyngeal consonants, [ʕ] and [ħ] (those links have sound files so you can hear). Eventually these weakened and were lost in Punic, but they were present in early stages.

It also had a series of "emphatic" consonants, which in Phoenician and Punic were probably ejective consonants, as they were in other early Semitic languages and still are in many South Semitic languages, such as those spoken in Ethiopia. These consonants sound to English speakers a bit harder and sharper than non-ejectives; the ejective [k'] sounds almost like a click (though it's not a true click consonant like those found in some Southern and Eastern African languages). Phoenician and Punic had three emphatics written /ṭ/, /ṣ/, and /q/, phonetically [t'], [ts'], and [k']. Eventually in Punic these seem to have lost their ejectiveness.

The consonants that are really tricky are the sibilants, or s-sounds. Phoenician and Punic had 4: aforementioned /ṣ/, and three others written /š/, /s/, and /z/. In Hebrew these were pronounced /sh/, /s/, and /z/, but originally in Proto-Semitic they were probably pronounced /s/, /ts/, and /dz/. Eventually Punic just merged them all, but in the middle period we don't know how they were pronounced. At some early stage of Phoenician they might have still been pronounced as in Proto-Semitic, since the Greeks borrowed the /s/ sign for /ks/ and the /š/ sign for /s/, which would otherwise be a strange choice, but we really don't know.

Our best guess as to vowels comes from foreign transcriptions, that is, texts in Punic or Phoenician that are written in an alphabet that shows vowels. An example would be a section of Plautus's Poenulus, a Roman play, which as a section in Punic. It's hard to interpret, though. For example, what do all those y's mean? Are they vowels like a German /ü/, as the Greek Y was, or do they represent something else?

The most similar sounding modern languages might be those South Semitic languages, like Tigrinya, that retain ejectives and pharyngeals, even though Tigrinya is almost as distant from Punic as it's possible to be while remaining a Semitic language and has several other phonetic differences (a completely different vowel system and a palatalized affricate series, for starters).