How did the government attempt to shape public opinion during the Cold War?

Propaganda and media.

One of the ways media was used was when media that was more appealing to the younger generation (children) such as comic books, textbooks and schooling was used to spread negative messages and enforce negative opinions of communism in the United States of America.

Education in America was a propaganda tool, using the ideas of democracy and freedom to present the Soviet Union as an enemy who did not agree with their views.

Posters were one of the most obvious examples of propaganda-google it and you'll easily find many different examples as well.