Conrad attended Clifton High School, where he was an All-state quarterback, while leading the team to back-to-back district championships in 1952 and 1953.
Conrad participated in the 1958 Chicago College All-Star Game and although he had never attempted a kick in college, he was able to make 4 field goals, 3 conversions, set the game scoring record with 15 points and also intercepted one pass in the 35–19 upset of the 1957 NFL Champion Detroit Lions.
On May 10, he was traded along with safety Dick Nolan to the Chicago Cardinals, in exchange for End Pat Summerall and halfback Lindon Crow.
He left as the seventh All-time pass receiver in the NFL with 418 receptions and had a string of 148 consecutive games played.
[4] On September 20, 1969, the Dallas Cowboys put injured wide receiver Bob Hayes on the "move list" and activated Conrad.