A matter of misunderstanding of those who don't read comics is that they are all superhero related. This isn't so. For ever genre of fiction you can find in a bookstore there is a comic that fits with it. There is not topic that cannot be explored in comics. My 1st recommendation is to think about what you like to normally and then look for that genre in comics. I am confident you will find something to meet your taste.
The following are some recommendations that might help you figure out what you like.
Understanding Comics by Scott McCloud is a fantastic place to begin your foray into comics. It is a comprehensive book that can be used as a road map through comics. Everything you wanted to know about the art form can be found here.
If you like myth/Shakespeare I would recommend the Sandman Series by Neil Gaiman. It is rich with literary and mythical allusions that add to the richness of the story. There is no other book that I know of that can deal with fantasy, horror, the macabre, mythology, theology and history in the beautifully human way that Sandman does.
Watchmen by Alan Moore is an excellent read, particularly if you are looking for the anti-superhero comic, every stereotypical them of superheroes is explored and ripped apart in this intelligent work of art. The movie was good, but can't hold a candle to the book.
A book that has been on my must read list for sometime is Maus by Art Spiegelman. It is a survivor's tail of the holocaust. You can't get further away from the traditional comics medium then that. It is also a winner of the Pulitzer Prize.
Here is also a list of writers and books that you might want to look into:
Writers:
Brian Michael Bendis - practically a writing god to Marvel
J. Michael Straczynski you know his writing whether you realize it or not
Kevin Smith Sir writes comics, can't get them in on time, but he does write them
Joss Whedon yeah from Buffy/Angel/Firefly
Frank Miller one of the best gritty storytellers of his generation
Jeff Loeb like JMS you know his work too
Grant Morrison excellent writer, sometimes hard to understand, but very good
Peter David - a writer similar to Loeb and Straczynski
Richard Donner wrote for Superman of course
Books:
Batman: Hush
John Constantine: Hellblazer
The Sin City Novels
The Dark Knight Returns
Annihilation
Marvel 1602
DMZ
The Killing Joke
The Long Halloween