en_tn_lite_do_not_use/act/19/08.md

Paul went into the synagogue and spoke boldly for three months

"Paul regularly attended the synagogue meetings for three months and spoke there boldly"

reasoning and persuading them

Probable meanings are 1) the words "reasoning" and "persuading" are a hendiadys that means "arguing persuasively" or 2) the words describe two separate activities, "giving them reasons to believe" and "convincing them" (see how you translated "reasoning" in Acts 17:2). (See: rc://en/ta/man/jit/figs-hendiadys)

about the kingdom of God

Here "kingdom" stands for God's rule as king. Alternate translation: "about God's rule as king" or "about how God would show himself as king" (See: rc://en/ta/man/jit/figs-metonymy)