In his youth, Drew showed great athletic talent. He won several medals for swimming in his elementary years, and later branched out to football, basketball and other sports. After graduating from Dunbar High School in 1922, Drew went to Amherst College on a sports scholarship. There, he distinguished himself on the track and football teams.
Drew completed his bachelor's degree at Amherst in 1926, but didn't have enough money to pursue his dream of attending medical school. He worked as a biology instructor and a coach for Morgan College, now Morgan State University, in Baltimore for two years. In 1928, he applied to medical schools and enrolled at McGill University in Montreal, Canada.
In 1938, Drew received a Rockefeller Fellowship to study at Columbia University and train at the Presbyterian Hospital in New York City. There, he continued his exploration of blood-related matters with John Scudder. Drew developed a method for processing and preserving blood plasma, or blood without cells. Plasma lasts much longer than whole blood, making it possible to be stored or "banked" for longer periods of time. He discovered that the plasma could be dried and then reconstituted when needed. His research served as the basis of his doctorate thesis, "Banked Blood," and he received his doctorate degree in 1940. Drew became the first African-American to earn this degree from Columbia.