The two boys' primary concerns initially seem to be whether they are on an island and if there are any grown-ups around.
“This is an island. At least I think it’s an island. That’s a reef out in the sea. Perhaps there aren’t any grownups anywhere.”
The fat boy looked startled.
“There was that pilot. But he wasn’t in the passenger cabin, he was up in front.”
Their worry about adults is a logical one. They are young and therefore need the guidance and assurance an adult could provide. Their concern about being on an island is also self-evident, later made clear by Piggy who mentions that rescue would be easier if they were on the mainland. Being on an island would make finding them much more difficult. Piggy also expresses the horrendous fact that they might live there until they die, if this were so.
Piggy is also concerned about the other boys:
“All them other kids,” the fat boy went on. “Some of them must have got out. They must have, mustn’t they?”
“We got to find the others. We got to do something.”
Piggy's concern indicates his care for others early in the novel and we find that he always expresses such concern throughout. It is ironic that, although he is the one most victimised by the others' name-calling, jeering and abuse, he genuinely wants them to all be sheltered and rescued.
A key concern of especially Piggy, is also whether they would be rescued.
“They’re all dead,” said Piggy, “an’ this is an island. Nobody don’t know we’re here. Your dad don’t know, nobody don’t know—”
His lips quivered and the spectacles were dimmed with mist.
“We may stay here till we die.”
Ralph seems confident that his father, who is in the Navy, would find them. Piggy is more insightful and realistic, though. He realises that the chances of rescue are indeed very slim, since no-one knows that they are on the island.
Piggy also expresses his worry about being teased and asks Ralph to not mention that he had been called 'Piggy' when at school. Sadly, Ralph finds this derogatory title quite funny and laughs at him. He later also shouts out this embarrassing nickname, making Piggy the object of everyone's fun. He however, does explain to Piggy that it is better than to be called, 'Fatty', when Piggy confronts him.
When they discover the conch, Piggy sees its value in calling the others together. He sees the importance of having a meeting and furthermore is the one who attempts to memorise the names of all the boys. He, in doing this, once again shows his concern for the welfare of others and the maintenance of order.
Finally, Ralph expresses concern about having a leader - one who can make decisions.
“Shut up,” said Ralph absently. He lifted the conch. “Seems to me we ought to have a chief to decide things.”
He is soon elected chief since he was the one who blew the conch, bringing all the boys together.