Born in Pavlovsky Posad, Russia, on 2 August 1934, Bykovsky was the son of Fyodor Fyodorovich Bykovsky and Klavdia Ivanova. He had an older sister named Margarita born three years earlier. When he was four years old, World War II began, forcing the family to move to Kuybyshev, and later again to Syzran, before moving back to near Moscow. By the age of 14, Bykovsky wanted to attend naval school; however, his father was not a proponent of this idea and encouraged him to stay at his school. A few days later Bykovsky attended a lecture on the Soviet Air Force Club which inspired him to pursue his dream of becoming a pilot. He began flight theory lessons when he was 16 at the Moscow City Aviation Club.

Bykovsky graduated from aviation school aged 18 and enrolled in the Kachinsk Military Aviation Academy. He graduated from the academy at 21 years old and received the rank of lieutenant. At 25 he became a jet fighter pilot and later became a pilot and parachute instructor with over 72 jumps by the time he began his cosmonaut training.