Except for the third use of the word "they" which refers to the people that Paul and Barnabas had led to the Lord, all the words "they" here refer to Paul and Barnabas.
"When Paul and Barnabas had appointed leaders in each new group of believers"
Possible meanings are 1) "Paul and Barnabas entrusted the elders they had appointed" or 2) "Paul and Barnabas entrusted the leaders and other believers"
Who "they" refers to depends on your choice for the meaning of "them" in the previous note (either elders or leaders and other believers).
"Word" here is a metonym for "message of God." AT: "(See: rc://en/ta/man/translate/figs-metonymy)
The phrase "went down" is used here because Attalia is lower in elevation than Perga.
This can be stated in the active form. AT: "where believers and leaders in Antioch had committed Paul and Barnabas to the grace of God" or "where the people of Antioch prayed that God would care for and protect Paul and Barnabas"