Peter is speaking to Jews, on their celebration of Pentecost, not to Jews and Gentiles. Acts Chapters One through Three make it clear. Peter does not speak of Christ dying for our sins, being buried and being raised on the third day for our sins, because he did not know this at that time. His message was a call to repentance for Israel, in order that the times of refreshing would come and Christ would return and set up His kingdom. He was preaching EXACTLY what Christ commissioned him to preach.
Also, they looked at everything from a Jewish perspective. "The Good News" was the Messiah had come and he would set up His Kingdom. They had no idea about a Spiritual Kingdom. Even after the Ascension of Jesus Christ they were looking for an immediate return once Israel repented and believed.