Attorney General Edwin Meese III took a large step toward installing his own conservative team yesterday as President Reagan announced he would nominate three people to key Justice Department positions.

Charles J. Cooper, a deputy assistant attorney general in the Civil Rights Division, will be nominated as assistant attorney general in charge of the Office of Legal Counsel. Cooper, 33, a graduate of the University of Alabama law school, is a former law clerk to Supreme Court Justice William H. Rehnquist.

Acting Solicitor General Charles Fried, who has been filing the administration's Supreme Court briefs since Rex E. Lee resigned last spring, will be nominated to be solicitor general. Fried, 50, is a longtime Harvard Law School professor who previously served as a special assistant to the attorney general.

Stephen J. Markman, 36, chief counsel of the Senate Judiciary subcommittee on the Constitution, headed by Orrin G. Hatch (R-Utah), will be named assistant attorney general for the Office of Legal Policy.

Cooper's nomination, to a job often described as the attorney general's lawyer, is likely to be the most controversial in the Senate. Cooper has been a key deputy to Assistant Attorney General William Bradford Reynolds and once helped defend the administration's effort to allow segregated schools to be exempt from federal taxes.

Reaction was quick among civil rights activists who helped defeat Reynolds' nomination to the department's No. 3 job. Ralph G. Neas, director of the Leadership Conference on Civil Rights, called Cooper "a right-wing ideologue who has sought to reverse the civil rights policies of every Republican and Democratic administration since Roosevelt."

William Robinson, director of the Lawyers Committee for Civil Rights Under Law, said that, "Cooper's views are virtually identical to those held by William Bradford Reynolds . . . extremely hostile to civil rights."

Fried is certain to be questioned about the friend-of-the-court brief he filed for the Justice Department in two pending Supreme Court cases, urging the court to overturn its landmark 1973 decision legalizing abortion.

Markman was tapped for the office that screens candidates for federal judgeships, after the administration decided not to nominate Christian activist Herbert E. Ellingwood, head of the Merit Systems Protection Board. In the meantime, Meese assistant Grover Rees, a former University of Texas law professor, has been spearheading the judicial selection process.