The companion who wrote much of Paul's missionary activities in the Acts of the Apostles is
Luke.
Luke is traditionally considered the author of both the Gospel of Luke and the Acts of the Apostles. Internal evidence within the text of Acts, particularly the "we" passages (where the narrative switches to using "we" instead of "they"), strongly suggests that the author was a traveling companion of Paul during some of his missionary journeys. These "we" passages indicate that the author was present for significant events in Paul's ministry.