I support Halx's recommendation of
Cowboy Bebop. But I didn't see the series; I saw the motion picture. But, man, what an awesome soundtrack.
Also, good introductions to anime would pretty much be anything produced by
Studio Ghibli. But Xerxys seems to be after violence and fighting, so I'm not sure there would be enough of that in these films. But...they are accessible and some I would call masterpieces.
As for violence, one only has to watch
Fist of the North Star to see how good it gets.