Born in Alice, Texas, United States, Curl was the son of a Methodist minister. Due to his fathers missionary work, his family moved several times within southern and southwestern Texas, and the elder Curl was involved in starting the San Antonio Medical Centers Methodist Hospital.  Curl attributes his interest in chemistry to a chemistry set he received as a nine-year-old, recalling that he ruined the finish on his mother's porcelain stove when nitric acid boiled over onto it.  He is a graduate of Thomas Jefferson High School in San Antonio, Texas.  His high school offered only one year of chemistry instruction, but in his senior year his chemistry teacher gave him special projects to work on.
Curl received a Bachelor of Science from Rice Institute (now Rice University) in 1954.  He was attracted to the reputation of both the schools academics and football team, and the fact that at the time it charged no tuition.  He earned his doctorate in chemistry from the University of California, Berkeley, in 1957. At Berkeley, he worked in the laboratory of Kenneth Pitzer, then dean of the college of chemistry, with whom he would become a lifelong collaborator.  Curls graduate research involved performing infrared spectroscopy to determine the bond angle of disiloxane.