Did the Apostle Paul speak Greek?
The apostle Paul definitely spoke Greek. He preached the gospel in Athens, read Greek literature and quoted Greek poets. Since he wrote his letters in Greek, his written Greek is extremely good, and he demonstrates a knowledge of Greek rhetoric. Also, when Paul quotes scripture, he does so from the Greek translation of the Septuagint (although he doesn't always get it word perfect, because he was relying on memory, or he may have known variant versions). He comes from Tarsus in Cilicia (now southern Turkey, on the Mediterranean), which was a very Hellenized city.
He may have spoken Aramaic. He refers to Peter in his letters as "Cephas", which is the original Aramaic form of "Rock", the nickname given to Simon by Jesus. Also, in 1 Corinthians 16.12 he uses the word "Maranatha", which is Aramaic for "Come, Lord." However, Paul describes himself as a Pharisee, a student of the Torah. This would require that he knew Hebrew, and probably Aramaic (the language in which the Talmud was ultimately written down). He seemed to be able to travel around Jerusalem and Judea with ease, so he probably knew the languages spoken there, namely Hebrew and Aramaic. For most of the 20th century it was usually thought that Hebrew in the 1st century was a dormant language used mainly in Jewish liturgies and rabbinical disputes, but more recent scholarship (largely based on the evolution of Hebrew in the Dead Sea Scrolls) suggests that it was actually a living spoken language at the time.
I would be surprised if Paul knew much Latin. Greek was the official language of the eastern Roman Empire, and there was an unusual gulf between Latin speakers and Greek speakers. Notably, when he wrote his Letter to the Romans he wrote in Greek, not Latin. Indeed, Greek remained the first language of the Church in Rome well into the 2nd century. The languages spoken in Galilee and Judea during the first century include the Semitic Aramaic and Hebrew languages as well as Greek, with Aramaic being the predominant language. Most scholars agree that during the early part of first century Aramaic was the mother tongue of virtually all natives of Galilee and Judea.
So in summary, we know from Scripture that Paul knew at least three languages: Aramaic, Hebrew and Greek. Beyond that, we know that God used Paul to reach many people whose language he did not know, through the gift of tongues. The question for us is, "Are we using our tongues, and the language(s) that we know, to tell forth the news that ‘Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures" (I Corinthians 15:3,4)?