All of the scriptures from Acts to Revelation with the word Synagogue in it that tell us what they were doing there.
Acts 13:14-41
They go to the synagogue.
He preaches to them Jesus
Act 14:1 And it came to pass in Iconium, that they went both together into the synagogue of the Jews, and so spake, that a great multitude both of the Jews and also of the Greeks believed.
Act 17:1 Now when they had passed through Amphipolis and Apollonia, they came to Thessalonica, where was a synagogue of the Jews:
Act 17:2 And Paul, as his manner was, went in unto them, and three sabbath days reasoned with them out of the scriptures,
Act 17:10 And the brethren immediately sent away Paul and Silas by night unto Berea: who coming [thither] went into the synagogue of the Jews.
Act 17:11 These were more noble than those in Thessalonica, in that
they received the word with all readiness of mind, and searched the scriptures daily, whether those things were so.
Acts 17:17
Act 17:16 Now while Paul waited for them at Athens, his spirit was stirred in him, when he saw the city wholly given to idolatry.
Act 17:17 Therefore disputed he in the synagogue with the Jews, and with the devout persons, and in the market daily with them that met with him
Act 18:4 And he reasoned in the synagogue every sabbath, and persuaded the Jews and the Greeks.
Act 18:19 And he came to Ephesus, and left them there: but he himself entered into the synagogue, and reasoned with the Jews.
Act 18:26 And he began to speak boldly in the synagogue: whom when Aquila and Priscilla had heard, they took him unto [them], and expounded unto him the way of God more perfectly.
Act 19:8 And he went into the synagogue, and spake boldly for the space of three months, disputing and persuading the things concerning the kingdom of God.
The same with the term Temple.
Act 2:46 And they, continuing daily with one accord in the temple, and breaking bread from house to house, did eat their meat with gladness and singleness of heart,
Act 2:47 Praising God, and having favour with all the people. And the Lord added to the church daily such as should be saved.
How does God add to the church daily? Through the teaching/preaching of the Gospel.
Act 5:19 But the angel of the Lord by night opened the prison doors, and brought them forth, and said,
Act 5:20 Go, stand and speak in the temple to the people all the words of this life.
They were instructed by an angel to speak in the temple of Jesus' life.
Act 5:21 And when they heard [that], they entered into the temple early in the morning, and taught. But the high priest came, and they that were with him, and called the council together, and all the senate of the children of Israel, and sent to the prison to have them brought.
Act 5:42 And daily in the temple, and in every house, they ceased not to teach and preach Jesus Christ.
Act 21:26 Then Paul took the men, and the next day purifying himself with them entered into the temple, to signify the accomplishment of the days of purification, until that an offering should be offered for every one of them.
Paul enters the temple to complete his Nazarite vow.
The scripture above is the one instance that tells what they were doing in the temple that doesn't say they were teaching.
But... it is interesting that it is here that the priests rise up against Paul.
Act 21:28 Crying out, Men of Israel, help:
This is the man, that teacheth all [men] every where against the people, and the law, and this place: and further brought Greeks also into the temple, and hath polluted this holy place.