What did Scout and Jem learn about segregation and the inequalities between blacks and whites in To Kill a Mockingbird?

From their experiences at Calpurnia's church and from
their own observations at the Tom Robinson trial, Jem and Scout discovered first-hand
the differences between the two worlds of the white man and black man. They were already
aware of the segregated living conditions, but they apparently had little contact with
Negroes other than with their housekeeper, Calpurnia. When she decides to take them to
her church, they are ready and willing to see a different side of Maycomb. Although they
are hassled by Lula as they enter the church and are aware of stares from the
congregation, they nevertheless are made to feel at home. They witness the poor
conditions of the church, the lack of hymnals (and literate people to read them), and
the repeated passing of the collection plate in order to gain donations for the Robinson
family.

At the trial, Jem and Scout sit in the separate
Negro section with Reverend Sykes. They here Tom referred to as "nigger" by both
witnesses and the prosecutor. They carefully examine the evidence presented by Atticus
and determine that Tom Robinson could not have harmed Mayella--but that Bob Ewell likely
did. It seems obvious to both children that the jury (though all white) will find Tom
innocent of the charges. When the guilty verdict is delivered, they are both aware that
the color barrier is the reason. As Atticus had previously hinted, no all-white jury in
Alabama can be expected to take the word of a black man over that of a white man--even
if the white man is Bob Ewell.