Anime Newbs

Death Note.

Hello, Welcome to Anilist owo! I have been to your anilist and have seen your profile and it's really cute. Have you made the banner yourself? if so it's really cool! As you don't have Code Geass in you plan to watch, that is the anime i recommend. It's realy really good and have one of the best plot twist in the whole anime world! I assure you it's an anime worth your time!

"Lelouch, do you know why snow is white? Because it's forgotten what color it's supposed to be." - C.C.

Uh so... there's few options..

another brainchild of One Punch Man's author, and it's dope:
Mob Psycho 100

who's the real monster? aka a story of a guy who finds himself torn between two worlds (if you watch and like it, consider watching Devilman Crybaby as well - the premise is pretty much identical, but the execution could hardly differ more):
Kiseijuu Sei no Kakuritsu

speaking of monsters, this thriller is imo somewhat of a classic you shouldn't miss (just beware it's psychological and doesn't exactly burst with action, though the atmosphere is gold):
Monster

and finally: Chihayafuru - just felt like throwing this one in, as it's not mentioned nearly as often as it'd deserve; it's a slice of life that relies heavily on its characters and their relationships and development, but it also offers some pretty suspenseful competitions, especially in the last couple of episodes

Death Parade

anilist.co/anime/21237-/

anilist.co/anime/102604 (2nd series is ongoing)

Mahou Shoujo MadokaMagica
(for the infamous ep. 3 plot twist)
or
Death Billiards
(because you've seen Death Parade but not this)

If you like thrillers and plot twists , you might really enjoy Psycho-pass
PSYCHOPASS
Also another one I watched quite recently which I highly recommend is Promised Neverland
Yakusoku no Neverland

ERASED

I think you would like Erased. It has those thriller elements you said interested you.

Also, obvious one, but SteinsGate

One of the best sci-fi thriller anime of all time.

Terror in Resonance : very interesting
Terror in Resonance

ERASED : a touching story <3 (I think you could start with this one)
ERASED

Kino's Journey : I'll let you discover ;)
Kinos Journey the Beautiful World the Animated Series

B: The Beginning : mystery and murder, everything I like
B The Beginning

Let me know if you'll try one of them!

Since you have Tokyo Godfathers in your favorites, I recommend you give Paprika, Perfect Blue, and Millenium Actress a shot because they're all done by Satoshi Kon.
Paprika
Perfect Blue
Millennium Actress
I also recommend Gankutsuou: The Count of Monte Cristo, which I'm currently watching and It's already one of my all-time favorite anime.

Gankutsuou The Count of Monte Cristo

One Outs

Could you name some non anime media you like? It make things easier to get a grasp of what you might enjoy ovo

Anything in my favorites

Samurai Flamenco
This will give you everything you want, and so much more. Maybe too much more, but too much is not a bad thing here.