First, it's really important to note when the narrator uses the first person point of view vs. the third person. The narrator struggles with living a fairly easy, middle class life during such atrocities of the Hitler era. Switching to third person is an easier way to deal with those atrocities. That being said, there are quite a few "main points" to mention in chapters 1-5.
First, it's important to note that Nelly had a happy, early childhood in the 1930s. Her parents were immersed in their retail business and weren't paying much attention to such matters as openings of concentration camps or the nixing of freedoms. It's hard to note poverty when you've achieved success.
Next, it's important to note the juxtaposition of Nelly's father being accused of consorting with Communists (easily brushed aside with bribes and gifts) vs. Nelly's parents beginning to vote more for the Nazi Party than for the Social Democrats.
The incident with Elvira and the Communist flag is important to note as well. Elvira (the family's maid) and her family were Communists, and they cried when the flags were burned. Nelly had to keep Elvira's tears a secret. Nelly begins to see that "things are all wrong," in that personal freedom is being taken away from so many people.
Another important point is the fact that Nelly wants to impress her teacher, Her Warsinski, who is a big supporter of Hitler. (The older Nelly, writing the memoir and speaking with her daughter, has great difficulty explaining why she had such positive feelings for him.) During this time, Nelly's family's business was booming, ... and they purchased their first picture of the Furher to hang up in support of Hitler.
Nelly's first experience with seeing adults be secretive about an affair is marred by the fact that one of the adulterers is Jewish. There is a definite feeling that one is doomed if participating in that religion.
You'll be wishing to go one of these day. And you'll be on your way. You'll become a wanderer. You'll have to want to go. You'll forget worrying about offspring. What does the Wandering Jew need a wife and child for?
A final main point in chapter four is that Nelly notices Dr. Leiner has his medical practice boycotted by the people of the town just because he is Jewish. We can connect with chapter five by noting that Uncle Emil is allowed to take over a candy factory, because it was abandoned by the Jewish owners.
Also in chapter five, Nelly struggles with her affection with her teacher, Herr Warsinski. Nelly always seems to fail in pleasing him, especially when she is compassionate to a family by giving them clothes for the winter, and Herr Warsinski tells the class about it only to have Nelly pelted with snowballs as a result.
Therefore, as you can see from the main points in chapters one through five, Nelly begins to grow up and enter school age. She is learning first-hand the every-day issues of living amid the influence of the Nazis.