The first time the President got their own plane was during the Second World War, when President Roosevelt ordered the creation of the Presidential Pilot Office in 1944.
The first Air Force One wasn't very impressive; it was a standard, propeller-driven plane which lasted for fifteen years. Eisenhower upgraded to a Boeing 707 in 1959, and John F. Kennedy was the first to have his own specially-designed plane built for him a few years later. Even that was really just a modified 707.