I was able to sort through a lot of shows thanks to a blizzard in New York City, which kept me inside and was not accompanied by losing electricity. My non-loss is your gain.
In the beginning, the show was a sort-of fanboy effort for UFO believers(1), a science versus supernatural thing where you knew the supernatural would win. But back then I only watched one episode, after it was already a full-blown phenomenon. I was unable to finish the one I watched back in the 1990s (2) but I watched it again this weekend. The description of it elsewhere - I consulted best-of lists because I didn't have time to watch all of them - wrote: "Hands down, this is the scariest episode of The X-Files. So twisted and demented that Fox has never reaired it", and that is true, which made it a bad test case for someone watching their first. It also had what I found was a recurring theme in the series; that big city life is awesome and rural people are inbred, slack-jawed yokels.(3) That episode didn't make my list, perhaps because the New York City mythology about Pennsylvania is tedious to people in Pennsylvania. (4)
None of the episodes on my list for newcomers who are sensible are the 'main' plot of UFO abductions that is the foundation of the show, because there is no way to do that well today, and in the 1990s it was still difficult to pull off credibly. Those stories just can't hold up in a post-Lost, post-Homeland, post-every-other-drama culture of today, where writing is just better. However, these stand-alone episodes are really good, and therefore basically timeless.
"The Post-Modern Prometheus" - Season 5, Episode 5 (Frankenstein! Bonus: One of multiple Pennsylvania slack-jawed yokels backdrops!)
"Monday" - Season 6, Episode 14 (Groundhog Day! Bonus: With bombs!)
"Bad Blood" - Season 5, Episode 12 (Vampires! Bonus: The fat kid from The Sandlot!)
"Clyde Bruckman's Final Repose" - Season 3, Episode 4 (Psychics! Bonus: Young Frankenstein!)
"Jose Chung's From Outer Space" - Season 3, Episode 20 (Self-parody! Bonus: I know you're not going to believe this but...!)
(1) There were a lot more of those then than there are now. When every phone has a high-definition camera, it gets harder to pull off grainy video. Also, no real school in America has a paranormal department any longer. In the show, science was always going to lose. In the real world...
(2) You can tell what year the X-Files episode is by the size of Scully's (she's the female lead) suit. In the early 1990s they were gigantic, Arsenio Hall-looking things. Her suits and hair gradually got less awful and by season 6 she no longer looked like Hillary Clinton with a red 'do. (His suits were bigger in the beginning too, but it is less jarring because men do a lot less fashion. I wear suits from 15 years ago and no one can tell - but if I wore one of those skinny, tight, Euro-weenie suits you would ask "It is still 2008?").
But you can also tell the period by the story-telling convention/cliches. There is a common shape-shifter cop-out in the show, which pretty much makes anything possible (because anything that happens can turn out to be a shapeshifter never caught and undo whole plot arcs). It seems there is a whole site dedicated to these kinds of story-telling tropes.
Their term for this particular cop-out, "Opening a Can of Clones", is inspired not just by X-Files, but was apparently happening all throughout the 1990s, when writers still didn't realize biology was really, really hard.
(3) Ditto. See "Twin Peaks" or any other show where the wholesome small town harbors a Dark Secret.
(4) The FBI is headquartered in D.C. and the show was filmed in Vancouver, but this thing absolutely screams NYC elitism about the rest of the U.S.