Of course, he could speak Hebrew and from such examples like preaching the gospel in Athens it seems that he had no problem speaking Greek. He also wanted to visit Rome, while there were no Jews living in Rome at that time, so does it mean that he could also speak Latin? Also he wanted to go to Spain. Does it mean that he could also speak Spanish? Was Spanish existent at that time?
I want to point out that my answer is going to be long since it is a long explanation.
The language of Apostle Paul (who was a Pharisee - Philippians 3:5) was Aramaic. Not Hebrew and Greek.
In Acts 1:19, Field of Blood was known to all the inhabitants of Jerusalem in their own language as Akeldama which is the transliteration of Aramaic words "Haqal Dama." (Source - Book "What do Jewish People think about Jesus?" by Dr. Michael Brown, Page 39)
According to First century Jewish Priest & Historian Josephus, Greek wasn't spoken in first century Israel. Josephus also points out the extreme rarity in terms of a Jew knowing Greek in first century AD.
"I have also taken a great deal of pains to obtain the learning of the Greeks, and understand the elements of the Greek language, although I have so long accustomed myself to speak our own tongue, that I cannot pronounce Greek with sufficient exactness; for our nation does not encourage those that learn the languages of many nations, and so adorn their discourses with the smoothness of their periods; because they look upon this sort of accomplishment as common, not only to all sorts of free-men, but to as many of the servants as please to learn them. But they give him the testimony of being a wise man who is fully acquainted with our laws, and is able to interpret their meaning; on which account, as there have been many who have done their endeavors with great patience to obtain this learning, there have yet hardly been so many as two or three that have succeeded therein, who were immediately well rewarded for their pains." —Antiquities of Jews XX, XI.
Dead Sea Scrolls Archaeologist Yigael Yadin also agrees with Josephus' testimony, pointing out that Aramaic was the lingua franca of this time period (Source - Book "Bar Kokhba: The rediscovery of the legendary hero of the last Jewish Revolt Against Imperial Rome" Page 234).
You may ask how is it possible that Apostle Paul didn't speak Greek since he was writing to people living in Greek regions. Apostle Paul was actually writing to Jews and Arameans who lived in Greek regions who spoke Aramaic.
Although Greek NT and the translations of Greek NT contradict the testimony of Josephus by saying that there were Greeks who communicated with Jesus and Apostle Paul, still Aramaic NT supports the testimony of Josephus by saying that they were either Aramaean(s) or Pagan(s).
Below are some examples.
Mark 7:26 (NIV Translation) - "The woman was a Greek, born in Syrian Phoenicia. She begged Jesus to drive the demon out of her daughter."
But in Mark 7:26 (Book "The Original Aramaic New Testament in Plain English", Page. 68) - "But she was a pagan woman from Phoenicia of Syria, and she was begging him to cast out the demon from her daughter."
John 7:35 (NIV) - "The Jews said to one another, "Where does this man intend to go that we cannot find him? Will he go where our people live scattered among the Greeks, and teach the Greeks."
But in John 7:35 (Book "The Original Aramaic New Testament in Plain English", Page. 153) - "The Judeans were saying among themselves, "Where is This Man prepared to go that we cannot be? Is He prepared to go teach the pagans?"".
Acts 16:1 (NIV) - "Paul came to Derbe and then to Lystra, where a disciple named Timothy lived, whose mother was Jewish and a believer but whose father was a Greek."
But in Acts 16:1 (Book "The Original Aramaic New Testament in Plain English", Page. 199) - "And he arrived at the city Derby and at Lystra, but a disciple was there whose name was Timotheus, son of a certain Jewess believer, and his father was an Aramaean."
Romans 1:16 (KJV) - "For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to every one that believeth; to the Jew first, and also to the Greek."
Romans 1:16 (Book "The Original Aramaic New Testament in Plain English", Page. 218) - "For I am not ashamed of The Gospel, because it is the power of God for the life of all who believe in it, whether of The Judeans first, or of the Aramaeans."
Acts 20:21 (NIV) - "I have declared to both Jews and Greeks that they must turn to God in repentance and have faith in our Lord Jesus."
Acts 20:21 (Book "The Original Aramaic New Testament in Plain English", Page. 206) - "While I was testifying to the Jews and to the Aramaeans about returning to God and the faith in Our Lord Yeshua The Messiah."
Acts 19:17 (KJV) - "And this was known to all the Jews and Greeks also dwelling at Ephesus; and fear fell on them all, and the name of the Lord Jesus was magnified."
Acts 19:17 (Book "The Original Aramaic New Testament in Plain English", Page. 204) - "And this became known to all the Jews and Aramaeans dwelling in Ephesaus and great fear fell upon all of them, and the name of Our Lord Yeshua The Messiah was exalted."
1 Corinthians 1:24(NIV) - "but to those whom God has called, both Jews and Greeks, Christ the power of God and the wisdom of God."
1 Corinthians 1:24 (Book "The Original Aramaic New Testament in Plain English", Page. 238) - "But to those who are called, Jews and Aramaeans, The Messiah is the power of God and the wisdom of God."
Galatians 2:3 (KJV) - "But neither Titus, who was with me, being a Greek, was compelled to be circumcised:"
Galatians 2:3 (Book "The Original Aramaic New Testament in Plain English", Page. 269) - "Even Titus, an Aramaean who was with me, was not compelled to be circumcised."
According to Dead Sea Scrolls archaeologist, Yigael Yadin, Aramaic was the language of Jews until Simon Bar Kokhba tried to revive Hebrew and make it as the official language of Jews during Bar Kokhba revolt (132-135 AD).
Yigael Yadin noticed the shift from Aramaic to Hebrew during the time of Bar Kokhba revolt.
In Book "Bar Kokhba: The rediscovery of the legendary hero of the last Jewish Revolt Against Imperial Rome" Yigael Yadin notes, "It is interesting that the earlier documents are written in Aramaic while the later ones are in Hebrew. Possibly the change was made by a special decree of Bar-Kokhba who wanted to restore Hebrew as the official language of the state"(page 181).
In Book "A Roadmap to the Heavens: An Anthropological Study of Hegemony among Priests, Sages, and Laymen (Judaism and Jewish Life)" by Sigalit Ben-Zion (Page 155), Yadin remarked: "it seems that this change came as a result of the order that was given by Bar Kokhba, who wanted to revive the Hebrew language and make it the official language of the state."
According to Book "Naming the Witch: Magic, Ideology, and Stereotype in the Ancient World" written by Kimberly B. Stratton (p. 232), Yadin suggests that Bar Kokhba was trying to revive Hebrew by decree as part of his messianic ideology.
You may ask what about mentioning Hebrew in NT. Whenever they saw Hebrew in first century Israel, they are referring to Hebrew tongue which was Aramaic.
John 19:17 (KJV) - "And he bearing his cross went forth into a place called the place of a skull, which is called in the Hebrew Golgotha."
John 19:13 (KJV) - "When Pilate therefore heard that saying, he brought Jesus forth, and sat down in the judgment seat in a place that is called the Pavement, but in the Hebrew, Gabbatha."
Words "Golgotha" and "Gabbatha" are Aramaic (Source - Book "The Words of Jesus in the Original Aramaic" Page. 114-115 by Stephen Andrew Missick & John 19:17 and John 19:13 of NIV). "tha" in "Golgotha" and "Gabbatha" is the Aramaic definite article on a feminine noun in an emphatic state (Source - Book "Introduction to Syriac" by Wheeler Thackston, Page 44).
For script tampering in Greek NT, check this link - http://en.metapedia.org/wiki/New_Testament
There are speculations in some quarters that Latin is extant - this was perhaps slumbering in the embryo of time, because of the four languages: Hebrew, Greek, Aramaic and Latin; Latin is the likeliest Paul could not speak regardless he was a Roman citizen
In Paul's day, Greek was still widely used in Rome, so knowledge of Latin was not crucial. Besides Hebrew and Greek, he most likely spoke Aramaic.
We know that He spoke Greek, Hebrew. May be Latin, but there is not reference about that. The Latin guess is made on the fact that he was Roman and the official langue of Rome was Latin.
Proof text for Hebrew:
Proof text for Greek :