James Brady, President Reagan's press secretary, Timothy J. McCarthy, a Secret Service agent, and Thomas K. Delahanty, a D.C. police officer, traveled vastly different routes to find themselves at the side of the president yesterday when they and he were shot.

Brady, 40, the man with the quick wit and ready rapport with reporters waited patiently for the press secretary's job while others were being considered. He is a veteran of Washington politics and public relations.

McCarthy, the 31-year-old son of a Chicago policeman, sought a Secret Service appointment from the time he was in college.

Delahanty, 45, a 17-year veteran of the D.C. police force now assigned to the K-9 Corps, found himself at the side of the president yesterday only because his police dog Kirk was ill.

Brady was the most grievously wounded. He was shot once in the head with a round from a .22-caliber pistol. Doctors said the bullet entered at the left side of his forehead and crossed to the right side of the brain before exiting. One doctor called it a "significant brain injury," another said that if Brady survives it is likely he will suffer brain damage. He under-went emergency surgery last night at George Washington University Hospital and was listed in critical condition.

Late last night, Sen. Charles H. Percy (R-Ill.) reported that the press secretary's wife Sarah visited him after the operation. Her husband recognized her and squeezed her hand, Percy quoted Mrs. Brady as saying, but was unable to talk to her because he had tubes in his mouth.

"Sarah was really very pleased," Percy said. "She said the reports she had been seeing on TV about him were more pessimistic than her conversations with his doctors."

Both McCarthy and Delahanty were described as being in stable condition.

McCarthy, the Secret Service agent, was shot in the chest. The bullet passed through his right lung, doing little damage, and lacerated the liver. His condition was upgraded from serious to good last night at George Washington University Hospital, where he was described as "doing extremely well."

Secret Service agent Dick Hartwig said that "from what I saw on video tape, it appears [McCarthy] stayed in between the line of fire" aimed at President Reagan.

Delahanty was shot in the lower left neck and the bullet lodged near his spinal column, officials at the Washington Hospital Center said. After a neurological examination was performed, a decision was made to not remove the bullet yesterday, but that decision is subject to continual review. His condition was changed from critical to serious yesterday afternoon and "the prognosis is good," hospital officials said.

D.C. Mayor Marion Barry talked with him there late yesterday afternoon and told reporters that Delahanty "said he wished he could have done more." Barry said Delahanty helped in apprehending the suspect, John W. Hinckley Jr.

Brady, a University of Illinois graduate and acknowledged master of press agentry, was working for John B. Connally at the beginning of last year's presidential campaign that propelled Ronal Reagan to the White House. When Brady was brought into the Reagan camp late in the Republican primary season, his role was ill-defined. He carried the title of director of public affairs and research, but it was never clear what that meant.

Brady endeared himself to the Reagan press corps with quips that sometimes got him in trouble with campaign officials. The most famous came when he shouted "Killer trees, killer trees," as the Reagan campaign plane flew over a forest fire in Louisiana. However, that reference to Reagan's comment that trees are a major source of air pollution did not amuse the campaign staff.

Brady is pudgy and round-faced and acknowledged those facts by referring to himself as "the bear." After graduating from Illinois, Brady taught government while working on a PhD for two years at Southern Illinois University. He first came to work in Washington in 1961 as a junior aide to the late Sen. Everett M. Dirksen.

Brady returned in 1973 and served in succession as an aide to James T. Lynn, when Lynn was Secretary of Housing and Urban Development and director of the Office of Management and Budget, and to Donal Rumsfeld when Rumsfeld was Secretary of Defense. Before joining the Connally campaign, Brady was an executive assistant to Sen. William V. Roth Jr. (R-Del.).

McCarthy was born in Chicago in 1949. He attended St. Leo High School and then the University of Illinois, where he graduated with a degree in finance. He won a letter on the football team as a walk-on, meaning he was not recruited and did not receive an athletic scholarship.

Tab Bennett, sports information director at Illinois and a teammate of McCarthy on Illinois teams in 1969 and 1970, remembered McCarthy as a "reserved type of guy, very detail oriented. People kind of looked up to him."

Bennett recalled that he was in the football training room one day when a "real stiff-necked guy" interviewed the trainer. The interviewer turned out to be an investigator checking McCarthy's qualifications for the Secret Service. "I never knew whether McCarthy made it or not, until today," said Bennett.

McCarthy's sister, Laurie, said McCarthy did heavy work on the Chicago docks while waiting for his Secret Service appointment, which came in 1972. He was assigned to the Chicago Field Office, then transferred to the presidential protective division in Washington in 1979.

His sister said, "Our dad was a Chicago cop for 35 years . . . that really inspired him, but he decided on the Secret Service . . . . "

McCarthy was assigned yesterday to the president's "traveling detail," one of the many assignments Secret Service agents regularly receive. McCarthy lives in Northern Virginia with his wife and two children. His salary is about $32,000 annually, the Secret Service said.

Delahanty is a K-9 Corps officer assigned to the D.C. police department's 3rd District (the downtown area), but his dog has been recovering from heartworms, a serious canine disorder, and thus Delahanty was available for other assignments.

He usually works with officer Winston Smith, who said yesterday that had Kirk not been sick, "Delahanty would have been off work like me" because he would have been assigned to a different work shift. Smith said he learned of the shooting from television. "I couldn't believe it," he said. "I saw his [Delahanty's] picture and I did not know if he was ducking or what. There was no doubt in my mind that it was my partner."

Delahanty's wife, Jean, learned of the incident the same way, she told friends at the Hospital Center. The Delahantys live in suburban Maryland and have no children.

Delahanty, a Navy veteran, was born in Pittsburgh, and joined the police force in September 1963. He is well-liked by his fellow officers, one of whom said yesterday that Delahanty "is one of the few guys who would work for you if you needed a day off." Delahanty's police record shows more than 30 commendations from citizens and superiors.