The first move from civilization to savagery was when the signal fire lost control and consumed much of the trees, killing the little boy with the birthmark. This showed a lack of foresight and responsiblity.
One of the big moves to savagery was when Jack killed the first pig and the boys become obsessed with the idea of slaughter with the ritual dance.
When Jack left the group and formed his own tribe, this signaled a fall towards savagery as the unity and coordination of the group is dissolving; especially because Jack's tribe is all about fun and killing rather than responsibility.
Perhaps the biggest fall to savagery is when Simon meets the "Lord of the Flies" and then is killed soon afterwards. It shows the evil within each of the boys that can['t be escaped.
One large symbol for the end of civilization is when the conch is smashed and then Piggy is killed, as the last sources of knowledge and order are destroyed.
Finally, the last of civilization is destroyed when the man hunt for Ralph begins, as the value of human life is next to nothing on the island.