General

Adding on to HolyDragon's list:

Episodes 1-2: a good introduction.
Episode 11 (Moonlight Sonata; 1 hour special): not Black Organization, but it's regarded as one of the best cases in its 700+ episode run
Episode 43: no spoilers for you, but this episode introduces two very important characters who are involved with Black Organization story arcs later on
Episode 76: the introduction of Kaito Kid, an important recurring character
Episode 304 (2 hour special): another one of the best episodes from the series. Nice introduction to the police force, some essential history, and an important exchange between Conan and Takagi that sets the type of relationship they'll have from this point onward
Episodes 667-668: Introducing Amuro
Episodes 671-674: Bourbon foreshadowing
Episodes 701-706: Bourbon arc climax

I haven't seen anything beyond 706 yet, so that's all I can add without guessing. [url=http://www.detectiveconanworld.com/wiki/Anime]This is currently the best episode guide available, so if there's anything else that looks interesting to you, go for it.[/url] The symbol(s) in the plot category tell you exactly what the episode is or isn't. So if the Black Organization arcs are the only things that intrigue you, watch all the episodes with a person symbol. TV original episodes can more or less be skipped.

@holydragon wow thanks man!

An old list of mine. It's outdated though. Aside from these. There are actually some really good cases though. A murder in a two sided mansion. A mansion where 8 detectives kill each other in an attempt to find the killer. A case where evidence suggests a man killed his newly wed wife. Turns out they're long lost twins and the woman, upon learning the truth and is pregnant, kills herself.

List Begin

Episode 128, Akemi Miyano dies.
Episode 129 Ai Haibara is introduced.

Episode 176-178, Conan and Ai find Gin's car on the street and Conan places a bug in it. He learns that the Black Organization plans to commit the murder at a party with famous people. The two head there, fail to stop the murderer whom is codenamed Pisco. A bunch of things happen and the status quo is resumed.

188-193- Conan returns as Jimmy for a while with a temporary antidote and has an incomplete date with Ran.

230-231 Introduction of Akai Shuichi

258-259 Introduction of James Black

277-278 Introduction of Jodie

284-285 Ran gets sick and regains a traumatic memory

286-288 Shinichi and Ran in New York meet Vermouth and Akai

307-308 Conan finds a game designer who works with the Black Organization. He then runs into Akai.

309-311 Conan tries to catch them.

345 Vermouth tries to kill Haibara.

385-387 Conan finds out the Black Organization's phone number

425 Black Organization tries to assassinate a political figure. They fail and Kir is taken in by the FBI.

429-430 Eisuke Hondo introduction

446-447 Foreshadow on Eisuke Hondo

462-465 Foreshadow on Eisuke Hondo

479 Foreshadow on Eisuke Hondo

484-485 Foreshadow on Eisuke Hondo

491-504 Black Organization tries to retrieve Kir.

507-511 Eisuke and Okiya

521-523 Shinichi's return

542-543 Okiya foreshadow

563-564 Akai foreshadow

578-581 Akai foreshadow
616-621 Shinichi and Ran romantic progress

646-650 Sera's introduction

656-657 A foreshadow on Okiya and Sera

List End