Yes. Play Batman. There is no "to play or not to play" there is only "play." Doesn't really matter where you start. Arkham City expands greatly on Arkham Asylum and understanding events in AA is not essential in AC (but they will be referenced anyway) chances are you'll end up playing Asylum after playing City anyway. Of course you should play both of them anyway. Oh, and if you play Arkham City, be sure to get the Catwoman DLC included somehow. It provides a more complete experience and best of all, it's fun! ^^
I rate the Arkham games among the greatest ever. They are so tight, so detailed, entertaining, and the greatest portrayal of Batman you'll see outside of a Christopher Nolan film. Yeah, I wasn't always a Batman fan but Nolan's films and the Arkham games made me into one.
Other than Batman, superheroes aren't exactly my thing. I'll enjoy the films before the comics, such as X-Men or Avengers.
Also, Fillion shows up in a surprisingly large number of video games as a voice artist. I think he and Adam Baldwin played minor characters in a Halo game, and Baldwin is unmistakable as the voice of Kal'Reeger in Mass Effect 2!