Adult males into Anime are weird.
(Cue outrage post by Haven in 3... 2... 1... )
Haha, that's the thing though... much like TV, movies, books... there are sub-genres.
Abundance of choice within each... kids anime is very different from young adult anime... which is very different from adult anime. There's some gritty, dark **** in anime. The only difference from an American drama show is... it's animated. Big deal, so is South Park.
I'd love to come up with a list of good "entry level" shows... things that are as big as "Cowboy Bebop" would probably be good... it's a space opera, very western feel to it... created by the legendary Shinichiro Watanabe. It was made in the late 90s, has incredible music, and a dark... serious story with plenty of gunplay and depth.
There are others that are just psychological horror... things definitely not for kids. Also some great mystery shows out there, just tons of good dramas all-around. Just have to get over the fact that... A. It's animated... and B. It may not always be available in English.