I'm just starting to get into anime, and would love some suggestions. They need to be stuff I can find on Netflix, please. So far I've watched:

All the Studio Ghibli films.... loved the artwork, plots were sometimes weak, but well worth watching just for the beauty of the backgrounds.

Cowboy Bebop: loved this

some Dragonball Z: got repetitious quickly, don't think I'm into the fighting as focus of the story stuff.

Tokyo Godfathers: LMAO

Saw NANA recommended on another thread and have watched the first disk and enjoyed it very much.

some Sailor Moon: too silly for me

Any suggestions? I'd really like to find stuff that isn't too sci-fi

Death Note is amazing. I love L, he's like a weirder Sherlock.
Black Butler (Oh my cute little Grell)
Attack on Titan (the feels will kill you, also violent as hell)
The first half of Sword Art Online (Don't watch the second half. Never watch the second half. It's hardly canon and just gets perverted)
Soul Eater (Never ignore it, watch the hell out of it!)

Try Detective Conan. Its pretty good. Look up "Detective Conan" to watch the Japanese version which has over 700 Episodes.
You can also watch the english dubbed version of the show called "Case Closed" but unfortunately you wont be able to finish the story because the dub was cancelled at 130 episodes. You can buy the entire seasons 1-5 of Case Closed on both XBL or PSN. They are also on Itunes and Google Play.

Seeing that you've watched all of Ghibli, all I can do is recommend "Grave of the Fireflies" if you enjoy tragedy. Waitaminute, you liked Cowboy Bebop but don't want sci-fi? How about Samurai Shamploo, kind of a retro hip-hop samurai spaghetti thing. Same people that did Cowboy Bebop.

Akira is pretty classic http://www.imdb.com/title/tt0094625/ - I know you said no sci-fi, but since you loved Cowboy Bebop, and I'd say that is more heavily sci-fi than this, you might enjoy it.
I got to see it at Sydney Opera House with Regurgitator playing a live score for it, was a pretty incredible experience.

I like the "Read Or Die" (R.O.D) series. There is a standalone OVA, then a small series that continues the story. http://www.imdb.com/title/tt0368197/?ref_=fn_al_tt_1 It's got great animation, great action, and great character development. Also, a personal favorite is Yu Yu Hakusho. It's a fighter anime, but unlike Dragonball, it also focuses on the characters themselves, and the plot moves faster than DB. There are four seasons to it, plus a movie. You can get each season for about $20, and with about 30 episodes per season, it's a great deal. Plus, Kurama is an awesome character. If you liked Bebop, then you may also want to check out Outlaw Star. It has about the same level of sci-fi, in that it takes place in the future and stuff, but it doesn't get too techy.

If you are interested in more mature fare (i.e. there may be occasional nudity, blood, or heavier themes like depression, murder, and suicide) I Highly recommend the movie Perfect Blue ( a J-Pop idol is being stalked), Ninja Scroll (another classic, like Akira), and, if you like having your mind played with slightly, Boogiepop Phantom.

Full Metal Alchemist: Brotherhood.
Trigun
Blue Gender

All of these anime are pretty good.

Thanks guys, my netflix list just got WAY longer! Death Note looks really interesting.... is that where Desecrate got his idea for the Sims? I can't wait for some of these to arrive in my mailbox. As far as the sci fi thing.... it's kind of like the fighting... it's ok if its sci fi, just not if the sci fi is the main focus of the whole story.

A lot of these are pretty damn good- definitely watch Samurai Champloo (you will get every song on the soundtrack stuck in your head, and that is a great thing ). Death Note is good- but for god's sake DON'T watch the dub. It's awful. It's also kind of the obligatory "gotta watch it" anime. If you like cartoony, dirty and stupid, I'd really recommend Panty and Stocking with Garterbelt (the dub is actually really good too- the jokes are more culturally relevant to a western audience- you can watch either, but I like the dub better- save for stockin's fuckin voice)

If you feel like embarking on a fucking saga, watch One Piece. There are 600+ episodes, but it is FREAKIN' INSANELY GOOD. It gets just plain nuts around the mid-250's, or the Water 7/ Enies Lobby arc. The show is hilarious and badass, and there's just tons of shit you don't see coming. It's got that whole "really fucking drawn out battle over 20 episodes" disease that DBZ's got, but once you get far enough in, you can easily just skip through the filler/flashbacks.

Just binge watched "Nana" and "Death Note". Thanks for that recommendation, jthm, it was awesome! There seems to be a sequel called "Death Note: ReLight", should I watch it? So often, sequels just totally screw up the series, has anybody checked this one out? (oh, and we'll have to wrestle for L, he's adorable, I want to adopt him and spoil him rotten)

Claymore is dark fantasy, I love that show.