A guide to interpreting Paul's letters is to examine how they are consistent with Torah and Tanakh, and not to wonder whether they are consistent.
Paul had an elite intellectual training and upbringing in the Pharisee tradition, based firmly on Torah and Tanakh. He approvingly witnessed the stoning to death of Stephen for his faith in Yeshua. Yeshua personally and dramatically appeared to him while he was in the act of persecuting followers of The Way (how early Messianic assemblies described themselves). His theology is deep, his understanding is vast, and his letters are meant to teach us and to help us put these doctrines into practice in our lives.
Shaul/Paulos does not contradict Torah or Tanakh. He would never do such a thing. Rather, he is trying to teach us the theology of Yeshua as the risen Messiah and how that relates to Torah, Tanakh, the Covenants, and also how Gentiles fit into the Kingdom of Elohim.
In Acts 16:3 we learn that he had his student Timothy circumcised. Does it make sense that he would then teach that circumcision was to be shunned?
In Acts 23:6 Shaul/Paulos declares:
My brothers, I am a Pharisee, the son of a Pharisee...
"I am," not, "I was". He did not cease being a Pharisee on the Damascus road or afterward. He calls the Sanhedrin "brothers" - they are Jews, he is a Jew. Shaul/Paulos did not give up his Jewish identity and neither did the other Shlichim.
Rav Shaul never changed his name to Paulos. As he was a Roman citizen of relatively high station in Judea Capta, he would naturally have his Hebrew name and also a name to use among foreigners. To his Jewish brothers he was Shaul, to the Romans and Gentiles he was Paulos. This is why he identifies himself as Paulos in his letters to the Asian and Greek assemblies; he apparently used his Greek name on his travels.
Wherever Shaul/Paulos travelled, he would stay in the homes of the sympathetic elders of the local synagogues. He would teach the Evangelion (the re-established Davidic Kingdom) of Yeshua HaMashiyach to the Jewish assembly in that city, and also to the Gentile believers among them. This is clearly enumerated in later chapters of Acts.
English translations today use words based on Greek, Latin or Old High German like "church", "bishop", "apostle", and "Christ" in the Brit Chadasha as opposed to "assembly", "elder", "shaliach", and "Messiah/Mashiyach". This difference tends to obscure the perception of what is being written about, making it seem less Jewish in nature. But if you read carefully, and mentally substitute these terms, it becomes more clear what Rav Shaul is teaching us. He is most certainly not replacing Judaism with a new religion, as the late 1st-century and 2nd-century 'church fathers' were all too eager to do.
After his experience on the Damascus road, Shaul had to rethink everything. He sought understanding in Torah, in Tanakh, in the teachings of Mashiyach. He learned how they complement, not contradict, each other. He learned how Yeshua HaMashiyach was the key to it all. He learned that the Gentiles were not to be excluded from the Kingdom of Elohim. He calls Luke's gospel "scripture" in 1 Timothy 5:18, quoting Luke 10:7.
Given the great depth of Rav Shaul's understanding, it really is no surprise that so many people, even well learned people in modern times, read Paul's letters and critically misunderstand them. Translation issues from (possibly Aramaic to) Greek to Latin to English, plus theological baggage from the anti-Nicene 'fathers' and Catholic and Protestant thinkers (the 'doctrines of men') make it even more difficult for the modern reader to comprehend Paul's messages to us. I feel that I myself only understand a small sliver of what Paul taught. Even after reading these other excellent answers, I feel I do not really comprehend the relationship of circumcision to the Gentiles. Perhaps even Mar Shimon, who was pejoratively nicknamed "the stone" (Peter, meaning dumb) by his peers, felt similarly. He wrote of Paul:
And count the patience of our Lord as salvation, just as our beloved brother Paul also wrote to you according to the wisdom given him, as he does in all his letters when he speaks in them of these matters. There are some things in them that are hard to understand, which the ignorant and unstable twist to their own destruction, as they do the other Scriptures. (2 Peter 3:15-16)
Paul's letters are, for the most part, master-classes in Messianic Pharisee theology. A person who has not studied Torah and Tanakh thoroughly already cannot expect to understand them properly, and they cannot be properly understood except within the context of Tanakh. And no one should expect to properly understand them when reading them one line at a time, or casually.