The biggest understandings Scout and Jem gain from their childhood experiences in To Kill a Mockingbird involve empathy and racial injustice.
Scout and Jem learn empathy throughout the book. In chapter three, Scout learns from Atticus that she shouldn't judge others until she "walks around in their skin." This comes after a particularly frustrating first day of school. Later in the book, Jem (and to some extent, Scout, too) learns empathy from his encounters with Mrs. Dubose. The children originally view Mrs. Dubose as a horrible, irredeemably mean old lady. By the time they learn of her death, and after a discussion with Atticus about her illness and morphine addiction, Jem and Scout come to understand the unique strength and humanity she actually had. Finally, the experience of Tom Robinson's trial gives an ultimate lesson in sympathy, showing the children how to "walk around" in the "skin" of Tom Robinson, Mayella Ewell, and a number of other Maycomb residents.
Scout and Jem also learn of the deep racial injustice that poisons their community. The experiences of the trial (and its outcome), visiting Calpurnia's church, witnessing the jailhouse mob, and more show the children how deeply divided Maycomb is.