Interesting topic. Let me start off by saying that I'm not going to touch on the new movie at all in this post. For one thing, knowing me, it's going to be a long post even WITHOUT that being brought up. For another, I haven't seen it yet, and don't want to judge it one way or another until I do. So! Onto the real post:
TOS is definitely not my fave. I do like certain eps, but honestly, I find a majority of it's eps to range from "Eh, it's ok" to downright awful. But for me, it comes down to "watchability". And a large part of what contributes to that is just an overall feel. Also, watchability is separate from significance.
Let me elaborate on all this. By significance, I mean how much impact a show had, how much respect I have for it, how good it was for it's time. And in that regard, TOS is certainly very significant. It was the first show of it's kind, it broke a lot of barriers, and of course, laid the groundwork for the franchise.
Watchability, though... not so much, for me. Now, others have mentioned that which show(s) you grew up with can have an impact on this, and I agree. I think that the fact that I grew up with TNG, and only saw TOS in the context of "this older show that came before TNG", has influenced what I like. But it's not just the fact that I watched TNG first, it's also about a certain overall style of storytelling, of TV show producing. Much of what makes TOS seem dated now comes (logically) from the fact that it was made in the 60's, and others in this thread have detailed those things quite well (especially JustKate and Gary7). The fact that TOS was made so long ago is an explanation for a lot of it, but it doesn't change the fact that those elements - even if they are present in the show for a reason - make it harder for me to watch the show. And that ties into something for me that relates not just to Trek, but to storytelling and TV shows in general: I like modern stuff better. This is especially true of certain aspects in particular; when it comes to TOS, the sexism is probably the one thing that I find the hardest to get past.
When it comes to stylistic elements (look of costumes and sets, look of the ships, style of background music), storytelling formula, and overall feel (which in TOS is very much "And this week, another strange new world!" *cue dramatic chord*), I personally find that seasons 3-7 of TNG and the entirety of DS9 are better in every way. I'm not talking about effects or film quality, I'm talking about characterization, story concepts, and production values. And I'm also not saying that these things in TOS are objectively bad, this is just about my personal viewpoint. But as I mentioned above, this expands beyond Trek, and is true of sci-fi in general, and really storytelling in general, in just about any medium. The campiness of TOS is very much one of the main things that makes it harder for me to watch, while for many people, that seems to be one of the main things they like about it.
As for the question (which I find a very interesting one) of what you think of when you say the words "Star Trek" and don't clarify which show you mean... I don't think of TOS. But I don't think of TNG or DS9, either. I don't think of any of the shows. I think of a setting. The name "Star Trek" calls to mind the universe that all of these stories are part of.
Having said that... while I don't think of any of the shows, per se, I do think of the 24th century before I think of the 23rd. For me, the 23rd century is like history or backstory. I still enjoy parts of it, and it's interesting to look back on, but the sub-section of Trek that always has and always will hold my interest the most is the late 24th-century. And yes, that includes Voyager; despite how flawed I found that series, I still liked it on the whole, just not nearly as much as TNG or (especially) DS9. Voyager had the potential to be much better than it was, and severely underused certain aspects (i.e., the Maquis/Starfleet tension, showing from week to week that heavy damage isn't just whisked away next ep like in TNG, etc), but it still was - at times - excellent, even if it had a lot of rocky times, as well. But going back to that concept of watchability... If you were to ask me, "Between TOS and Voyager, which show do you think better executed it's premise? Which show more effectively did what it set out to do?" I would say TOS. But if you asked me "Which show do you enjoy watching more? If you grabbed 7 random eps from each, which one has a more likely chance of entertaining you for those 5-6 hours?" I would say Voyager. TNG (barring the first two seasons... well, especially that abysmal first season), DS9 (only a handful of bad eps), and Voyager (LOTS of bad eps, and more than that, eps with poorly thought out concepts, but still, a large number of good ones, IMO) just have an overall feel, brought about by a combination of visual style, musical style, and writing style, that I just find much more engaging. This concept (which I hope I have presented in a way that makes at least a tiny bit of sense) is why I think that - by far - the finest production ever made involving the TOS characters and era is Star Trek VI.
Breaking down all the series, I'd say it goes like this for me:
DS9
TNG
Voyager
TOS (only a bit behind Voyager really... despite how much more "watchable" I find Voyager, TOS still gets points for it's significance and impact)
ENT (pretty far behind the pack... potential watchability is ruined by abysmal writing and an even greater failure to explore it's potential than Voyager. Also, it's one thing to not cling to canon like an oasis in a desert... it's another thing to just throw it out the window, and contradict well established aspects of a long-running franchise on a whim.)
All of that said, I don't look down on TOS or those who consider it the best Trek series. Everything I've said here relates to my own opinions; I may not get as much out of TOS as I do the later ones, but if someone else holds the opposite viewpoint, hey, that's fine with me. And while I find it less interesting than the later era, I do respect the fact that without the original series, there would BE no TNG, no DS9, no Voyager. It's important to remember that.
On the other hand, I think TOS laid a good foundation for fan elaboration so a lot of that unrealized potential does eventually see the light of day.
