Private 'lessons' with sensei, in glorious 3d pov.

What are some really serious or dark/depressing anime?

Oh there are lots.
These days there’s a lot of anime catered to more mature audiences - and many that deal with dark/gritty topics.
Even the “lighter” ones are bound to have at least a few serious monents - I guess it’s deemed necessary these days as far as catering to viewer preferences goes. Some examples: Parasyte: the Maxim, an anime about a boy whose right hand gets taken over by a monster known as a Parasyte.
Features a lot of death.
Explores in depth the dark side of human nature, with a special emphasis on arrogance, selfishness and greed. Tokyo Ghoul, thematically very similar to the above, but with a different premise. Death Note.
A show about a magical item known as a Death Note that can kill anybody when provided with a name and face.
Features tons of dubious morality and you get to see the main character slowly sink into madness just like in Breaking Bad. Psycho-Pass.
An anime about a world in which crime is anticipated and prevented by means of the Sibil System, a mind-reading device that maintains peace and order in society.
Again, lots of dubious morality and especially deals with fatalism, nihilism and determinism (the belief that free will is an illusion and the future is already predetermined.) Mirai Nikki / Future Diary.
An anime about a boy who gets a Future Diary that lets him predict the future.
Included here mainly because there’s a lot of death and you also get to see the originally innocent main character slowly corrupted by the necessity of killing. Code Geass.
This will always be a classic.
The protagonist is a textbook anti-hero, with messed-up ethics.
Features a lot of serious topics like death, killing and war. Elfen Lied.
Haven’t personally watched it but I think this fits the bill from what I’ve heard. Zankyou no Terror.
Pretty good anime that follows the lives of two former experimental test subjects who escaped and have turned to terrorism to get revenge.
Definitely very serious and mature, and features a lot of grey morality. Death Parade.
An anime that delves into the theme of mortality, and uses that as a backdrop to capture both the beauty and the horror of human nature. Garden of Words.
Deals a lot with the loneliness and awkwardness that is a normal part of human growth and maturation.
Bonus: has absolutely gorgeous background visuals. That should be enough to get you started.
Let me know if you want more. Edit: I missed out one that I consider to be one of the best recent animes: Puella Magi Madoka Magica. Don’t be fooled by the cutesy art.
It actually is an excellent deconstruction of the whole magical girl genre and deals with a lot of personal drama, death, and growth.
And it’s only 12 episodes! Definitely watch it.