What were the three most important achievements in the civil rights movement?

The answer to this question is ultimately subjective. The plight of black people before the Civil Rights Act was passed in 1964 meant that they were campaigning for more than three major changes—they were campaigning to be recognized as equals by their white compatriots in every area of society.
If we look at what was achieved by the Act itself, however, we can identify several key elements which could generally be agreed were hugely significant in the movement. First of all, the Civil Rights Act meant that segregation became illegal. Previously, under the Jim Crow laws, states had been allowed to segregate schools, bathrooms, and other public places so that black and white people could not mix. This was outlawed. Secondly, it became illegal for employers to discriminate against black people—and illegal for them to discriminate on grounds of gender, nationality, or religion.
Finally, in 1965, black people triumphed over the discriminatory legal practices that hindered their ability to vote with the passage of the Voting Rights Act. Suffrage has historically been one of the key cornerstones of most movements seeking equal rights, and this represented a significant victory for African Americans.