Pretty good list, from the ones I've seen. I'd prioritize Evangelion, but I'm a little biased since it happens to be my favorite Thing That Exists.

And don't be ashamed of not having watched Cowboy Bebop. I haven't finished it myself, though not out of lack of interest. :3

Add Revolutionary Girl Utena.

If you haven't already watched it, that is.

Oh, and Serial Experiments Lain.

And RahXephon

And Haibane Renmei

and Shinsekai Yori.

And... I think I'll stop there, for now.

Add Revolutionary Girl Utena.

 

If you haven't already watched it, that is.

 

Oh, and Serial Experiments Lain.

 

And RahXephon

 

And Haibane Renmei

 

and Shinsekai Yori.

 

And... I think I'll stop there, for now.

I already watched Lain, but I'll take the others into account.

I recommend Oreimo. If for nothing else, watch it for Kuroneko. <3

Gurren Lagann

Death Note

Ghost in the Shell

Cowboy Bebop (my shame is showing)

Porco Rosso

Princess Mononoke

High School of the Dead

Howl's Moving Castle 

I've seen all of those, I also recommend Graveyard of the Fireflies and the Sky Crawlers and I have been meaning to watch Wolf's Rain too

Sword Art Online
Don't even bother... it starts out good and then kind of staggers home in the second half.

Clannad and Clannad After Story

DOOOOOOOO IT! DOOOOOOO IT!
They are my all time favorite animes I bought the sound track.
 

Clannad and Clannad After Story

DOOOOOOOO IT! DOOOOOOO IT!

They are my all time favorite animes I bought the sound track.

 

They're on my crossed off list, man. I already watched them (and loved them). I even loved the controversial ending that WAS foreshadowed, since some people don't understand that apparently.

Madoka Magica - My brother has a boner for this show...
Princess Tutu - Really? REEEEEEEEAAAAALLY?
Higurashi No Naku Koro Ni (When They Cry) - Prepare for violence and confusing convoluted dimension travel... seriously everybody dies like 6 times.
Gurren Lagann - "WHO THE HELL DO YOU THINK WE ARE?!"
Attack On Titan - It is pretty good.
Kill La Kill - I'm going to watchthis as soon as I finish HenNeko.
Spice and Wolf - I need to watch this...
K-On! - I read the manga it was okay.
School Days (This will be fun  -_-) - He had it coming... that's all I'm sayin'
Neon Genesis Evangelion - I don't even want to watch this based on what I've seen...
Bamboo Blade - never even heard of it
Death Note - over rated.
Ouran High School Host Club - Read the manga it continues well on past the shows end.
Baccano! - Extremely confusing the first time through, but is amazing during the second watch.
Paranoia Agent - never seen it
Sword Art Online - *pushes off cliff*
Ghost Stories (English Dub specifically) - Isn't that the really terrible dubbed show?
Air - never seen it
Kanon 2006 - never seen it
Code Geass - DO IT DOITDOITDOTIODTIODIODDOIOODDODOOOOOIT!
Anohana - Angsty teens being angsty about angst... that's it...
Eden of the East - My sister liked this one.
Panty and Stocking with Garterbelt - watched about 6 episodes gave up
Ghost in the Shell - heard of it, didn't sound interesting
Black Lagoon - watch later list
Steins Gate - on my watch later list
Trigun - need to watch this one myself
Cowboy Bebop (my shame is showing) - Never watched it either... (shares the shame)
Girls Bravo! (I've got a very bad feeling about this one) - Yes feel bad
Wolf Children - Watch it and cry! Cry in a bottle and send it to me because tears are delicious!
Porco Rosso - My favorite Ghibli film
Laputa: Castle in the Sky -Great classic
The Wind Rises - I've heard of it never watched it
My Neighbor Totoro - Meh...
Princess Mononoke - Do it!
Tokyo Godfathers - Never heard of it
Revolutionary Girl Utena (Suggested) - Never heard of it
RahXephon (Suggested) - Never heard of it
Haibane Renmei (Suggested) - Never heard of it
Shinsekai Yori (Suggested) - Never heard of it
Oreimo (Suggested) - DOITDOITDOITDOTIDOTIDTODIDOITDOTIDOTIODOIT

My suggestions:
HenNeko
Hanamaru Youchien
Summer Wars (my favorite movie)

Also watch Gintama if you get around to it it has over 600 episodes though...
 

Edit: I bolded my comments because it was hard to read.
 

Madoka Magica - My brother has a boner for this show...

Princess Tutu - Really? REEEEEEEEAAAAALLY?

Higurashi No Naku Koro Ni (When They Cry) - Prepare for violence and confusing convoluted dimension travel... seriously everybody dies like 6 times.

Gurren Lagann - "WHO THE HELL DO YOU THINK WE ARE?!"

Attack On Titan - It is pretty good.

Kill La Kill - I'm going to watchthis as soon as I finish HenNeko.

Spice and Wolf - I need to watch this...

K-On! - I read the manga it was okay.

School Days (This will be fun  -_-) - He had it coming... that's all I'm sayin'

Neon Genesis Evangelion - I don't even want to watch this based on what I've seen...

Bamboo Blade - never even heard of it

Death Note - over rated.

Ouran High School Host Club - Read the manga it continues well on past the shows end.

Baccano! - Extremely confusing the first time through, but is amazing during the second watch.

Paranoia Agent - never seen it

Sword Art Online - *pushes off cliff*

Ghost Stories (English Dub specifically) - Isn't that the really terrible dubbed show?

Air - never seen it

Kanon 2006 - never seen it

Code Geass - DO IT DOITDOITDOTIODTIODIODDOIOODDODOOOOOIT!

Anohana - Angsty teens being angsty about angst... that's it...

Eden of the East - My sister liked this one.

Panty and Stocking with Garterbelt - watched about 6 episodes gave up

Ghost in the Shell - heard of it, didn't sound interesting

Black Lagoon - watch later list

Steins Gate - on my watch later list

Trigun - need to watch this one myself

Cowboy Bebop (my shame is showing) - Never watched it either... (shares the shame)

Girls Bravo! (I've got a very bad feeling about this one) - Yes feel bad

Wolf Children - Watch it and cry! Cry in a bottle and send it to me because tears are delicious!

Porco Rosso - My favorite Ghibli film

Laputa: Castle in the Sky -Great classic

The Wind Rises - I've heard of it never watched it

My Neighbor Totoro - Meh...

Princess Mononoke - Do it!

Tokyo Godfathers - Never heard of it

Revolutionary Girl Utena (Suggested) - Never heard of it

RahXephon (Suggested) - Never heard of it

Haibane Renmei (Suggested) - Never heard of it

Shinsekai Yori (Suggested) - Never heard of it

Oreimo (Suggested) - DOITDOITDOITDOTIDOTIDTODIDOITDOTIDOTIODOIT

 

My suggestions:

HenNeko

Hanamaru Youchien

Summer Wars (my favorite movie)

Also watch Gintama if you get around to it it has over 600 episodes though...

 

Edit: I bolded my comments because it was hard to read.

 

I don't that comment for Princess Tutu, don't judge a show based on its girlishness. *ahem* Other than that thank you for your suggestions, and yes I know the ending to School Days. and the Higurashi time loop.