Givens was born on January 5, 1930, in Quanah, Texas. As a child, his family called him Young Galen Givens. He took on extra courses when he was in high school, which allowed him to finish a year early. He worked in a grocery store and spent time cleaning cars to earn money for flying lessons and would hitchhike to Childress Municipal Airport to take them. His parents said he "never had an interest in anything but aviation". Givens earned his pilot's license in early 1946, and then performed a solo flight the day after he turned 16 in a Piper Cub. He graduated from Quanah High School in 1946. Givens was active in the Boy Scouts of America where he achieved its second highest rank, Life Scout.