Why did Peter and John go to the temple in Acts 3?
Peter and John went to the temple because the temple had not been closed and they were still practicing the Law, observing the required ninth-hour prayer. This was still an extension of the Old Testament program; they had not yet heard the Gospel of Grace, so they continued Jewish ritual.
Les explains that as chapter 2 ends and chapter 3 begins, nothing has changed; it is still an extension of the Old Testament program. Peter and John went up together to the temple, not to a church. The temple had not been closed down, and they were still practicing the Law, with no one telling them not to. They went at a required hour—the ninth hour, the hour of prayer. Les contrasts this with grace-age believers, who are not stipulated to meet at a particular hour for prayer. These men were still under the Jewish habit or ritual of going up to the temple at the ninth hour. While there, they encountered a lame man laid daily at the gate called Beautiful, begging alms. This setting fits the context that the Jews in Jerusalem had not yet heard the Gospel of Grace as we know it, and God was still dealing with Israel under the same circumstances that began with John the Baptist.
Now Peter and John went up together into the temple at the hour of prayer, being the ninth hour.