I could go on and on with recommendations, but I'll limit myself to a few right now.
For non-superhero books, I find that Oni Press has a really diverse line of narrative comics that a lot of people can get behind. An earlier commenter recommended Scott Pilgrim, and Oni published that.
1) Queen & Country (Oni) - This is currently on hiatus, but the books are all collected in "Definitive Editions" covering everything from the main title that has been published so far. There have been a few spin-off miniseries (Declassified, which covers back story of various characters) and a couple novels as well.
Queen & Country stars Tara Chace, an operative for the Special Operations Section of the UK's SIS (MI6). Greg Rucka wrote all of the main series, both novels, and two of the three Declassified series. This is in general a fascinating look at the politics and action of secret spy stuff.
2) The Adventures of Barry Ween, Boy Genius (Oni) - This one is written and illustrated by Judd Winick from the Real World: San Francisco. Don't let that deter you. There have been three miniseries about Barry, and they are all collected in "The Big Book of Barry Ween."
Barry Ween is a 10 year old genius stuck in a normal, midwestern life. He has the general misanthropic personality of a foul-mouthed 10 year old with the ability to change the world He does it while trying to stay relatively covert. This book is funny as hell and throws in some actual emotion and story telling, too.
3) Northwest Passage - Scott Chantler wrote and illustrated this series, which is collected in a single annotated volume. Set in 1755, this is a mix of American Western action adventure and British sea faring adventure that tells about a conflict over a settlement in the search for the Northwest Passage. Think of James Fenimore Cooper without the absolute boredom.
Like I said, I have a bunch more I could suggest, but I'll leave some for later.