The two most common themes within Lord of the Flies are the battle between civilization and savagery and the loss of innocence. Explain how this...

Those two themes are closely linked together because, as civilization falls apart for the boys, they lose their innocence at the same rate. Already in Chapter 1, the reader can see the threads of civilization already beginning to unwind from each other.  

First, the reader is shown that the only characters through the entire chapter are young boys between the ages of 6 and 12. There are no adults around. Furthermore, the island is completely uninhabited. There is no existing civilization or government structure. To the boys' credit, they do try to establish a form of civilization. They hold an election to nominate a chief, and Ralph is chosen as leader. Unfortunately Jack doesn't agree with the decision, and readers can already see a conflict developing between Ralph and Jack. 

Another piece of evidence for the loss of innocence in Chapter 1 is the trapped pig scene. It's amazing to me that Jack's initial response to the pig is to kill it. Granted, the boys are on an island and food is going to be an issue, but it isn't yet. The boys still believe that rescue is a good possibility. None of them have come to the realization that they might be on the island for quite some time; however, Jack's thoughts are immediately on violence and killing just for the sake of killing. There is no firm, civilized structure in place to curb his bloodthirstiness. Despite Jack's initial enthusiasm, he can't find the nerve to actually kill the pig. He still has some amount of innocence about him. 

They knew very well why he hadn’t: because of the enormity of the knife descending and cutting into living flesh; because of the unbearable blood.

The reader is only briefly comforted by Jack's apparent innocence, because he immediately vows to not hesitate to kill ever again. 

“I was going to,” said Jack. He was ahead of them, and they could not see his face. “I was choosing a place. Next time—! He snatched his knife out of the sheath and slammed it into a tree trunk. Next time there would be no mercy. He looked round fiercely, daring them to contradict.