Ye know (humeis oidate). Peter reminds his Gentile audience that the main facts concerning Jesus and the gospel were known to them. Note emphatic expression of humeis (you).
Beginning (arxamenos). The Textus Receptus has arxamenon (accusative), but the nominative is given by Aleph A B C D E H and is certainly correct. But it makes a decided anacoluthon. The accusative would agree with rhēma used in the sense of message or story as told by the disciples. The nominative does not agree with anything in the sentence. The same phrase occurs in Luk 23:5. Here is this aorist middle participle almost used like an adverb. See a similar loose use of arxamenos in the same sense by Peter in Act 1:22. The baptism of John is given as the terminus a quo. The story began with a skip to Galilee after the baptism just like the Gospel of Mark. This first message of Peter to the Gentiles (Act 10:37-44) corresponds in broad outline with Mark’s Gospel. Mark heard Peter preach many times and evidently planned his Gospel (the Roman Gospel) on this same model. There is in it nothing about the birth and childhood of Jesus nor about the intervening ministry supplied by John’s Gospel for the period (a year) between the baptism and the Galilean Ministry. Peter here presents an objective statement of the life, death, and resurrection of Jesus with proof from the Scriptures that he is the Messiah. It is a skilful presentation.