subarc, overlaps with BO storyRum Arc
For the sake of those who might want to get into or try to convince a friend to get into Detective Conan for the story but find the insanely high episode count and history of nearly 2 decades daunting, I've compiled a list of episodes that are actually important as well as listed the reasons someone may believe them to be important (so you can make your own judgments as you go as to whether you yourself really think you should have to watch them or not). There are tons of great "normal" cases throughout the series that naturally are left out, which are really the original point of the series, but if you just want to get around all of that and watch the series for just the overall story and characters, this might prove useful for that. Of course, there are still a ton of episodes, but cutting a good two-thirds of the series out should ease the pain of entry a bit. Now, I realize many others in the past have made feeble attempts at something similar, but I'm obviously more awesome than them. I plan to keep this reference updated periodically as the series progresses.
An updated note on more recent episodes: Since the series started, and especially in the last couple years, Aoyama has gotten infinitely better at weaving the story into the individual, unrelated cases. These days basically every case contains at least a bit of plot advancement, so plot advancement has become a lot more gradual and stretched out rather than happening in chunks scattered here and there. As a result, the more recent episodes pretty much all have some level of importance, except for anime originals, of course. Although, and I've stated as much before in the comments, as this list is about easing the barrier of entry, it is more the early episodes that really matter for trimming the fat. As you get closer and closer to completely caught up, it doesn't really matter all that much anymore anyway. In short, this list starts to lose meaning the further you get in the series and the closer to caught up to the present you are.
Another updated note on recent updated updates: Since DC has had some direct crossovers with the Magic Kaito manga over the years, yet those parts of the DC anime never had the Magic Kaito parts that gave them context animated as well, those crossovers have never really had their full and proper effect for anime only fans. However, now that Magic Kaito has finally received a proper anime adaptation, I have taken the liberty of working the timings of these all-important crossovers into where they fit in with their DC episode counterparts, making the crossovers whole once more. The Internet can thank me later. The only issue is that the Magic Kaito episodes are totally out of order, but the order really does not matter for that series like it does with Conan, as the series is completely episodic beyond pretty much the first episode, so you can ignore that and watch that series out of order all you want.
Note this uses the original Japanese episode numbers. The Funimation English dub version (aka "Case Closed" although the more recent official subs have adopted that name as well) stopped before it ever got to anything really big in the story anyway, so watching that would kinda defeat the purpose. Update: I went ahead and added the English dub episode numbers as well where applicaple in brackets. I still would recommend against using this list with the English dubbed version, especially considering that version was cancelled ages ago.
The movies are optional, seeing as they have nothing to do with any sort of plot from the series proper, but highly recommended based solely on outstanding quality. Note their placement in the list is based on when they were released in theaters in Japan relative to the original air dates of the episodes. This is because the movies tend to have spoiler references to things from the anime that came before the release of each particular movie, so watching a movie too early can be dangerous if you don't like spoilers.