Interesting thread! I've been meaning to get into some more shows. I've never been really captured by more than a handful while they were still around. I've really got to think about this one because I'm far more of a movie person and I'm not usually ready to commit to start watching a running series. Especially not nowadays when we're all swamped with an overload of everything being immediately available, so I'll try to avoid simply listing out the most recent shows I've really enjoyed.
In contrast to either of your lists, my choices are probably mostly anime. Possibly because I'm a fan of short (13-25 episode) series with a progressive story rather than plot of the week. There are exceptions, but those are usually what I'd consider most accessible.
In no particular order because I spent an hour trying to think of them, and might spend another one trying to sort them out.
The Simpsons - Particularly some of the early seasons. Might not need an explanation, and might even seem particularly vanilla on a top 10 list. But it's kind of crazy to watch episodes you haven't seen since you were 10 and being impressed with a lot of the wit that went flying over your head before.
Breaking Bad - Another one that might not need any elaboration because everyone and their grandma was fixated on this until its conclusion. The highest quality TV show I saw before this was probably Smallville or something because I was blown away by how impressively executed it was.
Daredevil - A new entry to this list, and what convinced me that small screen comic book based series aren't doomed to be cheesy or suffer low-budget looking effects. I've been burned out on Marvel superhero blockbusters for a while and this effectively killed them for me.
Shin Sekai Yori - And this is probably where the anime starts on this list. The scariest thing to me in the world is the concept of nihilism. In a world 1000 years in the future, a small, secluded group of humans with unique abilities start learning the truth about their history and the oppressive measures that had to be taken to prevent them from doing so. I'm not even sure the ending to this one was really anywhere near a high note; I was pretty disturbed by the unnatural artificiality of every aspect of not only the lives of the characters, but their very existence (within the story, not as a result of poor writing).
Neon Genesis Evangelion - Ditto for this one. If anyone had a passing interest in Anime at some point, they've probably seen this one. It's kind of funny going back and hearing misleadingly simplified synopses about how it's a mecha anime with giant robots fighting giant monsters. Eternally cynical about humanity and involving existential crises on an apocalyptic scale, there's probably been more head scratching as a result of this more than any other anime, or possibly even show, ever.
The Tatami Galaxy - An adaptation of a novel, it follows the protagonist through multiple parallel universes as he enrolls in different societies on his University's campus. Narrated at the speed of light and having episodes last less than 10 minutes, it moves at a breakneck pace as it portrays different possible realities for the protagonist as he tries to achieve the perfect campus life.
Paranoia Agent - Satoshi Kon is one of my favorite filmmakers of all time, and this series can stand up against the rest of his work. It revolves around a diverse set of characters who all have been affected in some way by a common assailant. Details are slowly revealed about why they have all been targeted. Things aren't as concrete as they appear to be, and there is a lot of blending between reality and imagination, in true Kon form.
Serial Experiments Lain - This entire show feels like a quiet trance. Released in the late 90s, it portrays the "Wired" as a sort of divine realm connecting everyone, and how the distinction between reality and the virtual world isn't so great.
Batman: The Animated Series - Setting new standards for what appears to be a kid's show, the show is excellently written. Portraying Batman in a noir, pulpy Gotham, it reinvented many of the ways Batman had been seen in the past and permanently affected how Batman and many of the other characters would be portrayed in future works.
Digimon Adventure - It's probably a childish choice, but I was 6 when this show first aired. It'll always have a degree of sentimental value for me, especially for being the first cartoon I've seen with a progressive storyline, an intense orchestral soundtrack, and surprisingly dramatic themes compared to what else was around at the time. I rewatched it recently for the first time in 15 years just for the sake of having a serious nostalgia trip. It held up better than I expected despite the rose-colored glasses. I'm just as excited as any other fan who grew up with it for the recently released series of movies serving as a sequel to the show.