Oh, I couldn't begin to tell you the first episode I ever saw because it was about 40 years ago when I first discovered Star Trek. So obviously I'm talking about TOS because the Trek movies and other series weren't even a twinkle in Roddenberry's eye at that time.
My primary memory is watching the show every night in syndication, on a UHF station on little TV in my parents' bedroom. I would lie on my parents bed and just watch and go through all these intense emotions as the stories unfolded. Of course I was only eight or nine back then, but compared to everything else that was on TV in the early 70s, Star Trek was so unique and different and I just loved it.