I've told the story before, but I can pinpoint it to 20 October 1966, when I was 15. I stumbled into the middle of "What Are Little Girls Made Of?" during a station break. First I saw a fake unconvincing cave set. Then some guy was running around in the cave. Some big ugly guy who looked pretty cool was looking for him. First guy pulls a phallic looking styrofoam rock from the ceiling and holds it at waist level (!). Ugly guy hears it and in a woman's voice, says "Captain Kirk! Where are you? I want to help you!"
Best episode description EVER.
The first ST episode I ever saw - that I can remember - was "Elementary, My Dear Data". I was a very small child, and I had no idea what was going on, but I liked it. It may have also contributed to my lifelong obsession with Sherlock Holmes. (To this day, I think Spiner did a better job playing Holmes than RDJ. But that's just me.)