Attack on Titan
I have never seen openings that better fit the show they belong to. Hands down my favorite anime openings. Just watch them if you haven't seen them. If you have, you know you want to watch them again. Guren no Yumiya and Jiyuu no Tsubasa - Linked Horizon
Soul Eater
These themes are great for wanting to understand the feel of the show. Resonance sets you up with the fast pace and the setting. Papermoon gives you more of the emotion behind it. And I Wanna Be throws the badassness of all of the characters at you. I listen to these on a regular basis. Resonance - TM Revolution
Monogatari
I love pretty much all of the songs from Bakemonogatari, Nisemonogatari, and the others. My favorite thing about them is that the openings are horribly misleading. They are all character intros so the show always seems so nice. The endings give you a better idea of how hectic and complicated the character relationships are.
staple stable - Chiwa Saito
Kimi no Shiranai Monogatari - supercell ft. Gazelle
BLEACH
All of these openings and endings are a great example of how different BLEACH is. The score to the show gives that in the first few minutes but these themes blend the comedy and lightheartedness (Asterisk and Tonight, Houkiboshi) with how heavy the show can be (After Dark, Velonica). Asterisk - Orange Range
Tonight, Tonight, Tonight - The Beat Crusaders
After Dark - Asian Kung-Fu Generation
Velonica - Aqua Timez
Houkiboshi - Youhna (has all the different versions)
Assorted Mix! Rewrite - Asian Kung Fu Generation - Opening #4 for Fullemtal Alchemist
It's basically impossible to find the actual opening so here's the song. This one gives a great feeling to the end of the original show; the desire to rewrite everything they had done. Almost nothing had gone the way they had wanted. The first show did a good job of creating something different than the manga by having the story just get away from the characters. The manga does a great job coming full circle while the first show just derailed. Rewrite was a great theme for the end of that trail. Again - Yui - Opening #1 for Fullmetal Alchemist: Brotherhood
Again gives a wonderful start to Brotherhood. The song itself is really awesome but the opening explores a lot of the characters and their relationships to each other. It also gives you a good glimpse into how different the animation actually is compared to the first show. My Story - The Cherry Coke$ - Opening for Giant Killing
This is one of my absolute favorites because when it starts, it's hard to think that it's Japanese. The song and the opening does a great job getting the viewer pumped for the show. It makes you one of the fans in the audience. Someone Else - Kana Asumi, Saki Fujita and Eri Kitamura - Opening for Working!!
Because why not? The opening is as crazy as the show is. It's ridiculously catchy and is just a fun time.