Whovians have it easy; the adventures of the good doctor from Gallifrey are fantasy in space. The only rules that apply are those of drama. Similarly, Mass Effect mostly nods to the science requisite to serve its actiony, genre-packed, Tolkienesque setting.
Star Trek was never smart enough to commit to either strict continuity or pure drama. There are hard rules to Trek, but drama, imagination, and morality tales take turns dragging those rules into a corner and working them over violently with spanners.The setting is packed with technologies and ideas that should change the universe's functions fundamentally. In Star Trek, the airplane would have been discovered, used to bomb Washington D.C., then thwarted by a sailing vessel and never been explored again in favor of slightly more ergonomic sails. It's a fucking mess.
None the less, dilithium, transporters, phasers, and antimatter form the cornerstones of the setting. There are also a number of principles which separate the citizens of the 24th Century from those of the modern day world.
The best way to tackle this--reasonably--is with a 10-episode crash course which covers the technologies, attitudes, and races of the setting. It sould show the best of Starfleet while not requiring everyone to watch every episode.
Good episodes don't necessarily count. Fifth season openers with extra drama and no aging lead cast members don't count. I love both halves of Best of Both Worlds, but only the first one covers the analytical nature, interpersonal conflict, and technobabble of the setting. Waltz is clearly, amazingly good, but it doesn't cover the points we need.
Some first-draft thoughts:
Errand of Mercy (TOS): It introduces The Romulans and there's significant discussion about Vulcans as well. Errand of Mercy is also about duty and showing that Trek villains can be nuanced people of honor. You can see that humas are still a bit racist, but those guys are generally regarded as crazies.
Disaster (TNG): Every member of the crew is thrown out of their comfort zone. Troi has to choose between the safety of some or trusting the lives of the entire ship to the rest of the crew. Generally, it involves more ingenuity and some technical discussion. It's also a hella good episode.
Destiny (Deep Space Nine): While the Defiant tries to work with Cardassians to create a telecommunications array through the wormhole, the Bajorans start freaking out over prophecies. Allies and enemies are oddly swapped and there's technobabble galore, intrigue, and clear conflicts between religion and science that are de rigueur for Deep Space Nine, but also representative of Star Trek's take in general.
Our Man Bashir (DS9): Put down the bottles and chains. A one-off parody episode set on the holodeck isn't the first thing you think of when you think "Essential Star Trek," but hang with me here. Holodeck episodes are a staple. Unless you're going to go with the TNG pilot episode "Encounter at Farpoint" or the one with Moriarty, then you're picking the least sickly of the residents at Madame Soapless' Orphanage and Medical Waste Dump. Second, there's some of the stretchiest stretches of Star Trek science in here, genuine character conflict, and if someone who doesn't like Star Trek watches Our Man Bashir and doesn't like it, then you know they're just being contrary.
For the Cause (DS9): Integrity. There's nothing else really significant about this episode except that Eddington sacrifices his career in Starfleet for industrial replicators, Sisko orders an investigation into his girlfriend for an unthinkable crime, and when all is said and done, Kassidy Yates turns herself in because it's the right thing to do. People do what is right, and they pay the price for it.
The Omega Directive (Voyager): I haven't seen this episode, but then I haven't seen a lot of Voyager. Apparently, Janeway has a job to do, sacrificing science for the safety of the galaxy. Seven of Nine contradicts her because she believes the science can be controlled. Again, I haven't watched it, but it seems like one of the episodes of Voyager that actually seems to have a good idea buried under all of the excessive amounts of Janeway and Seven, but nestled firmly above the atrophying layers of Torres and Kim. Also, they talk about the Borg, and they're somewhat important to the setting.
Again, this is just a first draft. I haven't seen a lot of Voyager and Enterprise, but I wouldn't be (too terribly, incredibly , unbelievably) surprised if there were some good episodes in there.