Discuss the Constitutional Amendments and Supreme Court rulings that have been the legal focus of the battle for civil rights. What roles did specific Civil Rights organizations, legal cases, and individuals play in enhancing equal access to voting rights, public accommodations, education, and employment?

The National Association for the Advancement of Colored People (NAACP) has played a key role since the 1950s on expanding voting rights and education to the Black American population. I will focus here on education.
The most important case regarding equal access to education is Brown v. Board of Education of Topeka, Kansas. The Brown case was a reversal of the 1896 Plessy v. Ferguson decision, which established the principle of "separate but equal" using the Fourteenth Amendment, as well as racial pseudoscience as the premise for the majority decision, written by Justice Henry Billings Brown:

The object of the [Fourteenth] amendment was undoubtedly to enforce the equality of the two races before the law, but in the nature of things it could not have been intended to abolish distinctions based upon color, or to endorse social, as distinguished from political, equality. . . . If one race be inferior to the other socially, the Constitution of the United States cannot put them upon the same plane.

In 1954, the Warren Court overturned the decision, also using the Fourteenth Amendment, by arguing that separate facilities are "inherently unequal" and, therefore, do not follow the amendment's premise of equal protection under the law.
Brown was a landmark case which resulted in the desegregation of schools. However, the South stalled in following the Supreme Court's order; as a result, Little Rock High School was not desegregated until September 1957.
There were other cases, however, that preceded Brown. One such example, Murray v. Maryland (1936), was taken to the Maryland Court of Appeals, which decided that the plaintiff, Donald Gaines Murray, should be granted a place at the University of Maryland's law school. Thurgood Marshall first took the case in 1933; he himself had not applied to the university's law school due to its racist policy. Marshall argued that Murray had demonstrated that he was as qualified as the white applicants and that "black" law schools were not of the same academic caliber as "white" law schools due to unequal allocation of resources. The latter argument would be one of the main reasons why black parents would later cooperate with efforts to desegregate white schools. The reason was simple: "black" schools were not given resources that were on par with those of "white" schools.
https://www.uscourts.gov/educational-resources/educational-activities/history-brown-v-board-education-re-enactment