I first saw Star Trek in re-run form when I was not more than five years old, and was hooked ever since. As a kid, my dad and I would watch it whenever FOX decided to air it, often weeknights at 7 pm, and now I wake up to it every morning on TV Land. It is probably my all-time favorite, and seeing its faults now as a discriminating adult simply endears the series to me even more.
The characters and their interaction are what I like best, as well as the sub-themes that ran with it, such as Spock's friendship with Kirk, and his unwilling but unavoidable bantering with McCoy. Scotty, Uhura, and Chekhov were equally charismatic, if sometimes underplayed. The villains were great fun, as was guessing who would be the first disposable character to get wacked on a landing expedition. It's easy now to criticize the special effects and sets, but we forget how exponentially technology has developed since the 1960s and we take for granted what we can now do with digital, which Rodenberry did not have at his disposal. Shatner's overwrought "acting" is simply icing on the cake, and great for a laugh; the paper-mache boulders and the crewmen literally hurling themselves across the bridge while under missile attack do bring a smile. All in all, if Star Trek were not among the greatest, there is no way it would have remained such an icon today.