The rest, as they say, is history.
Famously, the debut episode was not the first episode made, but the fifth. "MAN TRAP" about the Enterprise coming across a salt vampire, starred William Shatner and Leonard Nimoy. At this point, DeForest Kelley was only a co-star. The other featured players were James Doohan, George Takei, and Nichelle Nichols. Ensign Chekov and Nurse Chapel did not appear.
Here's a fan-made re-do of the opening and closing credits....it's awesome!
HAPPY BIRTHDAY, STAR TREK!