I'm a newbie and I'm looking for quality material. I'm open to any genre of anime. What do you recommend?

Cowboy Bebop*

One Piece

Kill la Kill

* highly recommended

The best one I've seen is Blue Gender : https://www.youtube.com/watch?v=KbwoBQTc23o

Don't know if it actually qualifies since it's produced in the US, but the Castlevania series was pretty good.

Cowboy Bebop

Fullmetal Alchemist: Brotherhood

Though it may be difficult to find a lot of good anime after watching that.

Berserk (the OLD series)

Claymore

Elfen Lied

Full Metal Alchemist + Brotherhood

Freezing

FATE STAY NIGHT/ZERO/UNLIMITED BLADEWORKS/HEAVEN'S FEEL

THE PHANTOM OF INFERNO

(new) THE RISING OF THE SHIELD HERO

Hellsing

VAMPIRE HUNTER D/BLOODLUST

Naruto Shipuuden

Yugi-Oh (try to locate uncensored Japanese dub English sub, anything else is censored heavily)

Blood +

And many others, that right now am too lazy to search. Ones in bold are masterpieces.

...guess, it's gonna end "What are your favorite titles?".
but hey, OP, maybe some hints on what plot \ setting features you'd like most? or you may get "Boku no Piko".

Sci-fi anime recommendations:

Legend of the Galactic Heroes - TV Series
All 5 Space Battleship Yamato Movies
Captain Harlock - 1978 TV Series
Galaxy Express 999 - TV Series from the 1970s
Space Adventure Cobra - TV Series

Sports Anime:

Ashita no Joe / Tomorrow's Joe TV Series (Boxing)
Hajime no Ippo / Fighting Spirit - All 3 TV Series (Boxing)
Captain Tsubasa - 1980s TV Series (Soccer / Football)

Shonen Jump:

Fist of the North Star TV Series
City Hunter TV Series
Kinnikuman - 1970s TV Series
Dr. Slump - 1980s TV Series

Giant Robot Anime:

Mobile Suit Gundam's 3 original movies (the ones with Amuro and Char Aznable)
Fang of the Sun Dougram - TV Series
Armored Trooper Votoms - TV Series

Also, any of Studio Ghibli's films, including and especially:

Spirited Away
Princess Mononoke
Nausicaa: Valley of the Wind
Laputa: Castle in the Sky

Uchuu Kyoudai -- the series tells a story of Mutta Nanba who has a chance to become an astronaut, a dream he gave up on long ago.

Kara no Kyoukai

I loved the Deathnote and Record of the Grandcrest Wars animes

Rose of Versailles. It's a 1970s anime that can be hard to track down, but it's a really well done take on the French Revolution, centered on a woman who becomes Marie Antoinette's royal bodyguard.

For someone new to anime without too much of a primer on what they like, I'll throw out:

The Twelve Kingdoms (Personal favorite, it's odd enough to start and becomes an enjoyable fantasy)
Code Geass
Mobile Suit Gundam The 08th MS Team (it only requires a little bit of a primer on the setting and is one of the better Gundam series. I did not like the first series very much. Otherwise, try Iron Blooded Orphans or Stardust Memory)
Sword of the Stranger (movie, great one at that)
Martian Successor Nadesico (this is my favorite show just to throw in every here and again)
91 Days (Good crime show that is still definitely an anime)
Darling in the Franxx (I dunno if this is actually a good one for someone new to this, but this was my favorite anime to come out in some time)
Someone mentioned it but I'll make special mention here that I really do love the first Fullmetal Alchemist series. Personally, I could not get into Brotherhood but everyone else I know has only seen the latter versus me with the former so it is also a good one. If you do give them a try, do it a favor and try watching the original one first.
Maybe the Full Metal Panic series too, but the first series is a harder sell than the incredibly funny Fumoffu and the great military action The Second Raid. If you do enjoy the first one, though, it does get nothing but better.
Another edit:
School Rumble is one of the funniest rom-com school shows and it's a good one to start with. It has a lot of characters, most of them are decently handled, and it gets you used to a lot of archetypes and tropes moving forward. On top of that, it is one of the funniest shows I have seen even after rewatching it a few times.
Super Dimension Fortress Macross is also a great mecha show where you can get some serious Top Gun vibes with the not F-14s. It is also an originator to a lot of mecha tropes and is a good one to start with.

https://www.gog.com/forum/general/anime_recommendations
There are a lot of recommendations waiting for you.