Daily Prayers were accompanied by the reciting of the Shema. While the Qumran community prayed three times a day, most Jews appear to have prayed twice a day. These prayers were either at the time of the morning and evening sacrifice or at dawn and sundown at whatever location the person happened to be at (they did not have to go to the synagogue to pray.) Home was the primary place of worship for the Jew. In addition to memorized prayers, individuals presented petitions to God for their own health and happiness.
The Eighteen Benedictions represent prayers used in public worship, although it is impossible to know for sure if they date to the pre-70 period. These prayers emphasize the attributes of God (his justice and mercy), his uniqueness, his willingness to forgive and to heal the sick as well as his control of the harvest. Sanders is doubtful first century Jews did any prayers (or hymns) in unison, but this cannot be certain since there is no evidence either way.
How often the average Jew studied the scripture is unclear. This may refer to simply going to the synagogue and heard the scripture read (especially for the non-educated who would not be able to read.) In addition, scrolls were expensive, only the incredibly wealthy would be able to own a scroll to study.
The Synagogue is of critical importance to the Jews of the first century. While we do not know when the synagogue was first used, we do know of synagogues dating to the first century (in the town of Gamla and one in Masada and the Herodian, likely built by Zealots long after Herod’s time.) Often synagogues were built over the site of an older building, accounting for the lack of first century archaeological remains. The synagogue at Tiberias was large enough to hold a crowd gathered to discuss the impending war (Life 277, 280, 290-303). We know from the Bible that both Jesus and Paul taught in synagogues regularly.
Philo describes the synagogue meeting which took place on Sabbath: a priest or elder would read from the scripture and comment on the text while people listened, then anyone who was moved to comment would do so. Usually they simply sat in silence and listened. Essenes were taught in the law every day, but more often on the Sabbath. The synagogue as designed with benches around the perimeter to encourage participation by all in attendance, as demonstrated in Mark 1:14-15, 6:1-5.
I am not sure Jesus challenges this institution of Second Temple Judaism. He is often described as participating in Synagogue discussions may have been asked on occasion to address the group as a teacher. But it is possible Jesus does subvert the normal practice of gathering to study Scripture at a Synagogue in other ways, but teaching on hillsides or other locations.