Fantasy is my #1; I will give almost anything a chance if it has strong fantasy elements. Post apocalyptic, superhero, alternate history, science fantasy, some supernatural, romance, and a few fandoms (especially Game of Thrones) are also likely to catch my eye.
The Wire, The Sopranos, True Detective and Generation Kill are all fantastic. My favourite shows in order I believe. The latter three are liable to switch and swap, but The Wire's been a personal favourite for years. I highly recommend it.
Now to be the obscure sci-fi nerd and suggest two aged classics.
Babylon 5: If you enjoy intelligently written science fiction, you'll enjoy this show. It's a space opera, which means it touches on many themes over many episodes with several arcs conjoined to a myth arc. First season's special effects did not age well. Beyond that, I have no criticism for this television masterpiece. If you like action packed television and don't like lengthy dialogue and philosophical debates, this probably won't entertain you.
Red Dwarf: Comedy? Check. British? Check. Commentary on the human condition with a surprisingly stellar soundtrack? Check. Like Babylon 5, it's intelligently written. Unlike Babylon 5, it's specifically episodic, not binding itself to a myth arc. Regardless, it's worth watching. Listen to the theme, and let nostalgia kick you in the spine.
I don't watch much TV, but I can contribute Person of Interest.
It's a modern day Scifi. I could explain what it's about, but the season 1 intro does it far better than I could.
It's very intelligently written, deals well with the ethical implications of building an all-seeing AI, knows it's technobabble very well, and never loses focus. The heart of the formula is the team saving (or stopping) "numbers" (called that because the Machine gives them a person's social security number when they're an iminent victim or perpetrator). Even when the team has to deal with extremely dangerous overarching threats like Decima or Vigilance, they still find time to go after the numbers. There are a few episodes that break the formula (such as If-Then-Else, M.I.A., and God Mode), but these are done just often enough (usually as season finales or otherwise climactic episodes) that they're a refreshing change of pace.
The characters are all excellent, and the show has enough humor that it's never excessively serious (mostly deadpan, as the characters are all master snarkers, and If-Then-Else shows us that even the Machine itself has a sense of humor).
I don't want to give too much away so I won't say anymore, but here are some of my favorite moments:
Urban Fantasy, High Fantasy, Epic Quest, Sci-Fi, Time Travel and World Hopping, Steampunk, Action/Adventure, Modern Drama, Mystery, Slice of Life, Romance, and many more.
Star Trek
Doctor Who (original or reboot)
Friends
X-files
(Mostly because these contain so many staple tropes of their genres, and for a few many people who haven't seen them have referenced them without knowing; Star Trek in particular is a big part of media culture)
*Video Game Highschool (If YouTube series counts)
-Doctor Who
-Fate Zero
-Breaking Bad
-Clannad and Clannad Afterstory
-Game of Thrones
-Cosmos (Neil deGrasse Tysons)
-Cosmos (Carl Sagans)
-Firefly
