What are the key moments in To Kill a Mockingbird and Lord of the Flies that show us the view of the world in the novels?

Rather than asking what are "the key moments" in To Kill a Mockingbird and Lord of the Flies that "show us the view of the world in the novels," we should probably ask what are "some" key moments in the texts. There are many scenes that reveal the authors' views of the world in both texts. 
Before identifying specific scenes that reveal these various views of the world, it's important to consider the texts in question. What do they have in common? How are they different? What are some of the underlying messages? It's fascinating how many similarities the two texts have. For instance, they both focus largely on the lives of children (Scout, Jem, & Dill in To Kill a Mockingbird and all of the school boys, including Jack, Ralph, Simon, and Piggy, in Lord of the Flies). Additionally, the adults in both texts are revealed as imperfect. In To Kill a Mockingbird, the jurors at Tom Robinson's court case are willing to declare him guilty because of racial prejudice, even when Atticus Finch has given solid evidence about his innocence. In Lord of the Flies, the adults (that we only hear about at the start of the novel and see, briefly, at the conclusion of the novel) are involved in a large scale war that is killing and harming many innocent people. This is, after all, the reason that the boys are being sent out of the country; the adults want to protect them.
Finally, both books were written at similar times: To Kill a Mockingbird was written in 1960 (though it took place in the 1930s) and Lord of the Flies was written in 1954. At this time, the world had recently experienced two World Wars, with World War II ending in 1945. Also, it's important that the world, specifically America (where Harper Lee was writing To Kill a Mockingbird), was experiencing major changes due to the Civil Rights movement. The historical contexts of these novels give added information that might impact the events and views the authors portray in the texts. Overall, both books are largely concerned with the nature of humanity: are people essentially good, or are they bent, twisted, and evil? Both authors explore the nature of humanity through children's points of view.
William Golding's Lord of the Flies reveals a negative view of humanity. As the adults wage their nuclear war, they send their children away to a place of safety. However, the plane is shot down and all adults onboard are killed. Now, the children fight for survival. As time progresses, what begins as fun and games ends in mass chaos and bloodshed. Even though they are children, many children are wounded and three children are dead by the end of the text. Just pages from the end of the novel, we see the destruction that the children were capable of. An officer arrives on the island to rescue the boys:

"Dumbly, Ralph shook his head. He turned a halfpace on the sand. A semicircle of little boys, their bodies streaked with colored clay, sharp sticks in their hands, were standing on the beach making no noise at all. 'Fun and games,' said the officer. The fire reached the coconut palms by the beach and swallowed them noisily. A flame, seemingly detached, swung like an acrobat and licked up the palm heads on the platform. The sky was black. The officer grinned cheerfully at Ralph. 'We saw your smoke. What have you been doing? Having a war or something?' Ralph nodded. The officer inspected the little scarecrow in front of him. The kid needed a bath, a haircut, a nose-wipe and a good deal of ointment.  “Nobody killed, I hope? Any dead bodies?” “Only two. And they’ve gone.” The officer leaned down and looked closely at Ralph. “Two? Killed?” Ralph nodded again. Behind him, the whole island was shuddering with flame. The officer knew, as a rule, when people were telling the truth. He whistled softly. 

The officer is initially cheerful. He asks if they've been "Having a war or something" and if anybody had been killed. His questions are likely playful, rather than serious, at first. After the boys state that two boys have been killed, the officer grows more serious and "look[s] closely at Ralph, trying to discern whether or not he is being honest. By the end of the text, we see the utter mess that a group of boys were able to make. We see how much destruction humanity is capable of, even children. Golding reveals the wickedness of humanity in his story.
Harper Lee's To Kill a Mockingbird shows a more positive view of humanity. While she shows many examples of people hurting one another (such as the jurors judging Tom Robinson unfairly and Bob Ewell lying to courts and abusing children), she also reveals goodness in other many other characters, such as Mrs. Dubose, who breaks her addiction to morphine despite immense personal suffering and Atticus Finch, who defends Tom Robinson even when it draws negative attention and violence from townspeople. Also, we see the goodness of Boo Radley, who originally is seen as a criminal but who later saves the children's lives.
One of the major themes in To Kill a Mockingbird is moral instruction. Atticus often sits with Jem and Scout, teaching them about society and showing them how to behave rightly. For instance, he teaches them not to harm the innocent:

"I'd rather you shot at tin cans in the back yard, but I know you'll go after birds. Shoot all the bluejays you want, if you can hit 'em, but remember it's a sin to kill a mockingbird."

Through this short lesson, Atticus is reminding the children that mockingbirds don't harm anyone; they simply make beautiful sounds for everyone to enjoy. Bluejays, however, are known to be mean birds, knocking eggs out of other bird's nests and causing destruction. Atticus reminds the children not to harm the innocent, a lesson that later applies to people, and not only birds. Overall, To Kill a Mockingbird reveals a much more positive view of human nature than Lord of the Flies does.