I sometimes talk a lot, so, bear with me.
Dragon Ball Z:
For starters, made and animated in the late 80's by none other than Toei studios. Crazy amounts of fighting, and if you watch the original Japanese version, you'll get all the cussing and weird inappropriate behavior as well. It's the kind of series you really don't have to pay attention to fully to enjoy, but it's considered a true classic by anime fans. If you like this, try watching the precursor, Dragon Ball. It follows the main character from Z in his childhood days.
Late 90's anime, featuring a space western theme, complete with guns and a main character who's just as corny as he is kindhearted. You can kind of think of this series as a 'spiritual brother' to Adventure Time.
Fullmetal Alchemist Brotherhood:
Not as action-packed as the others, but a good series all around. Follows two brothers on a quest to perfect their alchemy and revive their dead mother. The story expands greatly from there, leading them down paths they never knew existed, fighting wars and, eventually, a god-like being. I chose Brotherhood over the original, as it follows the manga (comic, Japanese style) more closely and has a slightly darker theme overall. If you like this though, try the first run of the series.
Fist of the North Star:
An absolute classic. Remember all those Bruce Lee movies you've seen over the years? Of course you do. Now, imagine them taken up a few notches and add a pair of thick eyebrows and a few head explosions. That's what this is.
Vampires, guns, and Nazis. The original series (simply Hellsing, not Hellsing Ultimate) had a better tone to it overall. Though, if you liked the first run of it, you might like the second series as well.
Short series that revolves around a small middle school and a strange curse. Some say it starts off slow in the first few episodes, but it quickly picks up around episode 3-4. By the 11th and 12th episode, you'll be wondering if anyone will be left alive.
Another short series focusing on a lone survivor of a middle school massacre, who is then locked up for the crime he didn't commit in the unique, theme park-esque Deadman Wonderland prison. I didn't like the ending much, but, I put it up here just for all the insane stuff that goes on.
Grave of the Fireflies:
Not an anime series, nor is it really horror or violent based. It's a movie that takes place just as WWII is ending, and revolves around a brother and sister and the will to survive. A wonderful, sad story.
So, just as a warning, if you watch this and you're too tired and or hungover, the voiceovers will give you a headache. Other than that, it's a fun, nonsensical series that pokes fun at itself and the genre in general.
Another movie. If you're into feeling like you just took a hit of acid, this is for you.
Gigantor and Astro Boy:
Two different series, but up here for the same reason. Because.
Transformers and Voltron:
Ok, not really an anime series, but, c'mon. They're awesome.
Not quite a 'true' mecha/robot series, but still a favorite. So Speed Racer retro. So fun.
I must admit, I'm not an expert on this area. But other series that are considered to be great within this genre are: Neon Genesis Evangelion, Patlabor, Gundam (in it's various forms), Macross and Robotech.
Uh...that's all I can think of for the moment. I hope that's good though.
If you want copies, just let me know. And if you want any more suggestions, you'll know where to find me.