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 I believe them to be important (so you can make your own judgements 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