What lessons do Scout and Jem learn from their encounter with Mrs. Dubose?

Jem has a blow-out of his temper when Mrs. Dubose, of whom he is terrified, enrages him by insulting Atticus (of whom she is actually a friend, with a mutual, shared respect between them). As a consequence, Jem is compelled to go to her for over a month (a week being added to the originally appointed month) to read to her while she is in a very, very ill condition. During these encounters, Jem learns courage; to control his emotions; to pursue the right course of action even when it is distasteful (or even appalling); and to turn a blind eye to what is distressing. After these encounters Jem learns, the hard way and much against his will, to respect someone with whom he deeply disagrees; to acknowledge and value courage; and to embrace as worthy a different point of view.

Mrs. Dubose was a lady of fiercely staunch Old South habit, expectation and upbringing as illustrated by her tirades to Scout: “Don’t you say hey to me, you ugly girl! You say good afternoon, Mrs. Dubose!” During Jem's quite unsettling reading-aloud encounters with Mrs. Dubose, Jem has to learn physical courage just to mount her "steep steps." As he confides in Atticus, Jem finds the Dubose home "all dark and creepy ... [with] shadows and things on the ceiling...." To forebear in the face of Mrs. Dubose' symptoms of suffering, Jem has to learn moral courage, as Atticus tried to teach him before, by receiving her insults with bold calm and emotional tranquility:

he would gaze at Mrs. Dubose with a face devoid of resentment. ... [H]e had cultivated an expression of polite and detached interest, which he would present to her in answer to her most blood-curdling inventions.

After the encounters with Mrs. Dubose, Jem learned about all-encompassing courage when he understood from Atticus the determination, strength and deeply rooted courage that Mrs. Dubose needed to call up in order to break her morphine habit so that she might pass from her life and "leave this world beholden to nothing and nobody." This lesson changes Jem in a concrete way that manifests in his attitudes and behavior toward Scout, much to her confusion and dismay: "Jem hollered, 'It’s time you started bein‘ a girl and acting right!'"

Ironically, Jem's lessons to control his emotions (not have a fit of camellia attacking while reading to Mrs. Dubose), to do what's right (steadfastly read aloud to her as his punishment), to turn a blind eye (not be distracted from his task by the appalling symptoms of her suffering) all stand him in good stead and allow him to be of help to younger Scout during Tom Robinson's trial. Although the experiences of that year, from the camellia beheading to the readings to the camellia-in-a-candy-box to the Robinson trial, took youthful Jem a bit further away from young Scout, coming-of-age Jem was bolstered in later troubles by the trials he coped with in Mrs. Dubose' sickroom.