You might take a look at Code Geass (and if you like that, you might also check out Durarara, which is less action, but more drama). Of course, there's always the classics like Dragon Ball Z and Yu Yu Hakusho, if you like the heavy shounen type of stuff. Hunter x Hunter got a remake in 2011 that's currently ongoing (though you can still watch the original series, if you'd prefer). It's not an anime, but you should really watch Avatar: The Last Airbender. If you like the idea of more realism in your shows, you could look into Hajime no Ippo, which is a boxing anime.
One Piece, Fairy Tail and Bleach are all really popular, but they may be too "over the top" for your taste.
Unfortunately, I'm not really versed in action/adventure anime. I watch a lot of Slice of Life and Psychological stuff.