1. It's good to see Peter get out of Jerusalem finally. He and John went up to Samaria in Acts 8 because they had not received the Holy Spirit. At the end of Acts 9, we fine Peter in the broader region of Judea and Samaria some more, fulfilling stage 2 of Acts 1:8.
2. Peter heals a paralyzed man (Aeneas) and raises a woman from the dead (Tabitha). This arguably is meant to show that, with the power of the Holy Spirit, Peter does things that Jesus did. A trajectory is being constructed which points toward today. We can do things that Paul did, which were things that Peter did, which were things that Jesus did, because of the power of the Holy Spirit in us.
3. Peter stays in the home of Simon, a tanner. Lots of dead animals around there. We're busting out of the Levitical purity rules. Acts doesn't make an explicit deal out of it, but the subtext is that Jesus has abolished the purity rules of the OT.
4. Acts 10 really brings this point home. Cornelius is a full blown Gentile, a Roman centurion (commander of 100), apparently from Italy. The gospel has spread to Samaritans. It has spread to a eunuch and perhaps a tanner. It is slowly breaking down purity and ethnic barriers. No it fully jumps to the Gentiles.
5. Perhaps most of the earliest converts to Christian Judaism were from "God-fearers." These were Gentiles who worshiped Israel's God but who had not gone all the way to be circumcised (for men). God singles him out here because of his prayers and gifts to the poor (10:4).
6. Peter sees a vision three times of God giving him a net full of unclean foods to eat. Peter refuses the first two times because he is following the purity laws of Leviticus. God corrects him by saying not to call anything impure that he has cleansed.
It's an obvious indication to Peter that he should go to the house of Cornelius, an unclean Gentile. We today might want to debate whether Peter is really violating the OT to go there, but Acts assumes that he is (10:28). The main point of the story is clearly that Gentiles can be saved too.
7. God stops all debate over whether Gentiles can be included before it even starts. God fills them with the Holy Spirit and that's that. This is the obvious conclusion they reach in Jerusalem after Peter reports back (11:17-18). Peter is criticized at first when he returns to tell this story. But the Gentiles even speak in tongues--probably an indication that their experience of the Spirit is in no way inferior to that of the Jews. They are baptized.
8. Peter's sermon to Cornelius and his men also confirms this point. God does not show favoritism--he receives people from every nation when they fear him and do good (10:34). sermon has similar elements to the other sermons in Acts: 1) God anointed Jesus with the Holy Spirit and power in his ministry, 2) they killed him but God raised him from the dead, 3) we are chosen witnesses to his resurrection. Other elements include the prophets foretelling and the final judgment coming.
9. There are some curious aspects to this story. For one, we find Peter in Galatians 2 not eating with Gentile Christians who are already believers. That is several years after this incident. How could the Peter of this story have any questions about eating with Gentile Christians long after this boundary has been crossed?
Paul never thinks of Peter as apostle to the Gentiles, as that Galatians 2 story points out. Acts seems to want to authenticate the Gentile mission by connecting it to Peter, with Paul as a good little boy following orders. It is at least possible that Acts has given us a more "decent and orderly" version than it was at least experienced at the time. Acts is not a videotape, but "apologetic history."