In verse 32, the story shifts from Saul to a new part of the story about Peter.
Verse 31 is a statement that gives an update on the church's growth.
This is the first use of the singular "church" to refer to more than one local congregation. Here it refers to all the believers in all the groups throughout Israel.
"lived peacefully." This means the persecution that started with the murder of Stephen was finished.
The agent was either God or the Holy Spirit. This can be stated in active form. AT: "God helped them grow" or "the Holy Spirit built them up" (See: rc://en/ta/man/translate/figs-activepassive)
"Walking" here is an idiom for "living." AT: "living in obedience to the Lord" or "continuing to honor the Lord." (See: rc://en/ta/man/translate/figs-idiom).
"with the Holy Spirit strengthening and encouraging them"
This phrase is used to mark a new part of the story. (See: rc://en/ta/man/translate/writing-newevent)
This is an generalization for Peter's visiting the believers in many places in the region of Judea, Galilee, and Samaria. (See: rc://en/ta/man/translate/figs-hyperbole)
The phrase "came down" is used here because Lydda is lower in elevation than the other places where he was traveling.
Lydda is a city located about 18 kilometers southeast of Joppa. This city was called Lod in the Old Testament and in modern Israel.