Acts 6...first mention of a synagogue.
Acts 6:9 But opposition arose from members of the Synagogue of the Freed Slaves (as it was called), composed of Cyrenians, Alexandrians and people from Cilicia and the province of Asia. They argued with Stephen,
We all know what happened to Stephen at the end of his confrontation with the Synagogue members.
Acts 13, 14 & 17 have Paul gaining believers from the Synagogues. He at times had to talk to them 'after' or took them 'aside'.
Even by the end of Acts 13 he is stating this:
"It was necessary that God's word be spoken first to you. But since you are rejecting it and are judging yourselves unworthy of eternal life - why, we're turning to the Goyim! 47 For that is what ADONAI has ordered us to do: 'I have set you as a light for the Goyim, to be for deliverance to the ends of the earth.' " 48 The Gentiles were very happy to hear this. They honored the message about the Lord, and as many as had been appointed to eternal life came to trust. 49 And the message about the Lord was carried throughout the whole region. 50 But the unbelieving Jews stirred up the women 'God-fearers' of high social standing and the leading men of the city, and they organized persecution against Sha'ul and Bar-Nabba and expelled them from their district.
He went to the synagogue but they expelled him.
And by the next chapter they fair even worse for having gone into a synagogue.
Acts 14:1 In Iconium the same thing happened - they went into the synagogue and spoke in such a way that a large number of both Jews and Greeks came to trust. 2 But the Jews who would not be persuaded stirred up the Gentiles and poisoned their minds against the brothers. 3 Therefore, Sha'ul and Bar-Nabba remained for a long time, speaking boldly about the Lord, who bore witness to the message about his love and kindness by enabling them to perform signs and miracles. 4 However, the people of the city were divided - some sided with the unbelieving Jews, others with the emissaries. 5 Eventually the unbelievers, both Jews and Gentiles, together with their leaders, made a move to mistreat the emissaries, even to stone them; 6 but they learned of it and escaped to Lystra and Derbe, towns in Lycaonia, and to the surrounding country, 7 where they continued proclaiming the Good News.
They went to the synagogue and the outcome was the members wanted to stone them.
Acts 17 proves no better:
1 After passing through Amphipolis and Apollonia, Sha'ul and Sila came to Thessalonica, where there was a synagogue. 2 According to his usual practice, Sha'ul went in; and on three Shabbats he gave them drashes from the Tanakh, 3 explaining and proving that the Messiah had to suffer and rise again from the dead, and that "this Yeshua whom I am proclaiming to you is the Messiah." 4 Some of the Jews were persuaded and threw in their lot with Sha'ul and Sila, as did a great many of the Greek men who were "God-fearers," and not a few of the leading women. 5 But the unbelieving Jews grew jealous; so they got together some vicious men from the riffraff hanging around in the market square, collected a crowd and started a riot in the city. They attacked Jason's house, hoping to bring Sha'ul and Sila out to the mob. 6 But when they didn't find them, they dragged Jason and some other brothers before the city authorities and shouted, "These men who have turned the whole world upside down have come here too! 7 And Jason has let them stay in his home! All of them are defying the decrees of the Emperor; because they assert that there is another king, Yeshua!" 8 Their words threw the crowd and the authorities into a turmoil, 9 so that only after Jason and the others had posted bond did they let them go.
10 But as soon as night fell, the brothers sent Sha'ul and Sila off to Berea. As soon as they arrived, they went to the synagogue. 11 Now the people here were of nobler character than the ones in Thessalonica; they eagerly welcomed the message, checking the Tanakh every day to see if the things Sha'ul was saying were true. 12 Many of them came to trust, as did a number of prominent Greek women and not a few Greek men. 13 But when the unbelieving Jews of Thessalonica learned that the word of God had been proclaimed by Sha'ul in Berea as well, they went there too to make trouble and agitate the crowds. 14 The brothers sent Sha'ul away at once to go down to the seacoast, while Sila and Timothy stayed behind.
They went to the synagogue but the outcome was riotous mobs against them and their Mission, Messiah.
Acts 18
4 Sha'ul also began carrying on discussions every Shabbat in the synagogue, where he tried to convince both Jews and Greeks. 5 But after Sila and Timothy arrived from Macedonia, Sha'ul felt pressed by the urgency of the message and testified in depth to the Jews that Yeshua is the Messiah. 6 However when they set themselves against him and began hurling insults,
he shook out his clothes and said to them, "Your blood be on your own heads! For my part, I am clean; from now on, I will go to the Goyim!" 7 So he left them and went into the home of a "God-fearer" named Titius Justus, whose house was right next door to the synagogue.
He went to the synagogue and they turned against him, and he said he will no longer go to the synagogues. Probably for fear of death. Just like Yehsua said.
Acts 19
8 Sha'ul went into the synagogue; and for three months he spoke out boldly, engaging in dialogue and trying to persuade people about the Kingdom of God. 9 But some began hardening themselves and refusing to listen; and when these started defaming the Way before the whole synagogue, Sha'ul withdrew, took the talmidim with him, and commenced holding daily dialogues in Tyrannus's yeshivah
Went to the synagogue and eventually the unbelieving Jews started defaming the Mission before the whole synagogue. So Paul uses the Yeshiva next door. Spite? Or divine convienence?
And now in Acts 22 a reference to what Paul did to all believers who attended synagogue before his conversion.
Acts 22:19 I said, 'Lord, they know themselves that in every synagogue I used to imprison and flog those who trusted in you;
Again, to imply that the leaders where leading new gentiles into these synagogues is implausable.
Last mention of a synagogue in Acts
Acts 26:11 Often I went from one synagogue to another, punishing them and trying to make them blaspheme; and in my wild fury against them, I even went so far as to persecute them in cities outside the country.
Yet, the apostles were leading believers into these synagogues?
So yes, Shaul and the emisaries did enter synagogues after Yeshua told them what would happen. And that is just what happened. Unbelieving Jews countered the Mission Paul was given.
Annnnnnnd so on and so on to today.