What does Luke 24:12 mean?
It appears that Peter—and John—go to the tomb because of Mary Magdalene's first report. She was with the women who had come to bring spices to cover Jesus' body. When they saw the stone rolled away from the entrance and the tomb empty, she left immediately. She ran to Peter and John to tell them, "They have taken the Lord out of the tomb, and we do not know where they have laid him" (John 20:1–2).Peter and John run to the tomb; John surges ahead of Peter, but Peter goes into the tomb first. They see the grave clothes but no corpse. The face cloth, which had been coiled around Jesus' head, is folded neatly. At this point, John says, he "saw and believed," but he doesn't say what he believed. That Jesus' body was gone? Or that Jesus had risen from the dead? We don't know (John 20:3–9).
While Peter and John return to the men, Mary stays at the tomb. At that point, she finally sees the angels and Jesus. He sends her back to the disciples to tell them He's alive (John 20:10–18). Around the same time, the women who had stayed when Mary ran off also arrive to tell the men they have seen Jesus (Matthew 28:8–10).
When the text says "home," it doesn't mean Peter returned to Capernaum in Galilee. The eleven remaining apostles and other disciples have been hiding in a locked room for fear of the Jewish leaders (John 20:19). There's every reason to think Peter returns to this room and is there when Jesus appears later that night (Luke 24:36).
This verse seems to challenge this proposed timeline, but it doesn't really. Luke just tells the women's whole story, before transitioning to the men with Peter. It's unclear why Luke doesn't mention John; perhaps Theophilus only knew Peter.