When did the Apollo 1 program start?

Apollo 1 wasn't a program, it was a mission scheduled to launch in February, 1967. However, the capsule caught fire on January 28, 1967 and killed the crew.

The first Apollo launched on October 11, 1968.

Apollo actually started during the Kennedy administration as a means of achieving his goal of "Landing a man on the moon in this decade and returning him safely to the earth."