This show has almost a religious significance to me and of course countless others.
Growing up in the 70's, my first TV crush was on Veena (the Cage/Menagerie). The first girl I kissed was a blue-eyed blonde, and later I married a blue-eyed blonde, well, three of them, to be perfectly honest. I think it's Veena's fault. I never did score with a green Orion slave girl, though, despite what they say about them.
While there are many stinker episodes of Star Trek, there are also many masterpieces. "City on the Edge of Forever" is my favorite, for its emotional impact. Who could not fall in love with a young Joan Collins? Another noteworthy episode is "The Doomsday Machine," which is interesting because of its unique musical score, with different instruments/styles for each character. You can find a YouTube video about it, by user shemvonschroeck. I think the most important episode is "Mirror, Mirror", for its (probable) social commentary on American imperialism, the Vietnam War, FBI Director J. Edgar Hoover's spying on members of Congress, and the CIA's use of torture and assassinations in the 60's. We are living in the "Mirror, Mirror" universe. The Utopian Star Trek universe is not our own.
If you like Star Trek, then of course there are the official sequels/spin-offs, but also you should check out Star Trek Continues. And if you like "Mirror, Mirror", then you should check out its Star Trek Continues counterpart, "Fairest of Them All".
I'm very torn over the original Star Trek series. I'm too young for it; I grew up in the 1980s with the original cast films - which I loved and still do - but my real adoration for Trek began with The Next Generation and then especially Deep Space Nine. The original Star Trek is a very different show from any of that, and I have to look at it with a different mindset to try and appreciate it. I don't have any nostalgia colouring my view.
To put it bluntly, it's horribly dated and oftentimes difficult to watch or enjoy. It's campy and looks very cheap. BUT, it's saving grace is how good the actors, writing and characters are. They brought the show to life, and at points made it a complete joy. For the time it was made it did incredible things with progressive storytelling and strong special effects.
I'm never going to love the original series, or even really get it. It's not my Star Trek, but it has its place in history and that can't be denied. I feel it got it chance to shine when it moved into the film format (conversely, TNG and the rest of the franchise worked far better on TV than as films).