I've decided to do openings first because, well, it's probably easier. Maybe. Maybe not.
I might redo this if I change my mind. Anyway, this is currently in no order. I might end up editing it so that it has an order. Also, sorry about the size of the videos. You can full screen it.
Edit of that: I created a list, listing all these openings in order of my favorites (basically, what I said in the mini-paragraph above this). Here's the URL if you didn't want to look it up: http://www.anime-planet.com/users/NicoNicoDesu/lists/34836
Higurashi No Naku Koro Ni OP
Lucky Star OP (Some people may not want to watch this...)
Soul Eater OP 1 (I'm pretty sure the song is Resonance by T.M. Revolution)
Shiki OP 1
Umineko No Naku Koro Ni OP (This video extends it. The anime opening ends at around 1:30)
Deadman Wonderland OP
Mirai Nikki OP 1
D. Gray-Man OP 1 (I guess the song is called Innocent Sorrow...)
Darker Than Black OP 1 (The song is called Howling)
Death Note OP 1 (I'm sorry if you prefer the second opening)
Elfen Lied OP (The song is called Lilium, and it is absolutely gorgeous. Also, this contains nudity)
Good Witch of the West: Astrea Testament OP (Like a better version of the Lucky Star OP)
Serial Experiments Lain OP
Cowboy Bebop OP (Song: Tank. 3, 2, 1, let's jam)
Ghost Hunt OP 1
Togainu no Chi OP (I think the song is Rose-Hip Bullet by Granrodeo...)
Kuroshitsuji (Black Butler) OP 1
11eyes OP (Song name: Arrival of Tears)
Well, maybe I'll add more later.