I'd be interested in hearing about some good comics that you've read that aren't based around super heroes. You mentioned Batman but I've never been too big of a fan of Batman, not that I wouldn't be willing to give it a try. What would you recommend to someone who's new to comics? One comic that I've heard praise from quite a bit is the Sandman series. I actually own The Dream Hunters which is apparently part of the Sandman Series (I actually bought this because Yoshitaka Amano did the artwork for the book) but that's not a comic.
Like I said, I haven't actually got round to readin em. FYI though, Sandman, like many others, may not be a superhero comic, but is still set in the same "universe" as the likes of Batman, The Flash, The Demon, etc. That said though, any "outside influences" are typically kept to a minimum, and the series is predominantly self-contained. One series I'm lookin forward to in particular is Lucifer, which is a spin-off from Vertigo's Sandman. Much like Hellblazer (the John Constantine comics) though, it sits on my hard drive unread, a trend it has followed for four years now.
Batman's always a great place to start though. What's made the character and comics so popular over the years is the sheer flexibility of Gotham and it's supporting cast. Not to mention the fact he's been written and drawn by a who's who of the industry's elite (and yes, that includes Marvel Mogul and good friend of Bob Kane, Stan Lee) If there's a specific type of story you want to read, it's almost guaranteed that it's in the Batcave's archives.
That said though, it's usually best to start with a Frank Miller book (be it Year One or The Dark Knight Returns), before going through the early chronological motions with The Man Who Laughs, The Long Halloween, and finishing up with the brilliantly handled Dark Victory. Then it's a case of cherry-picking classics like Matt Wagner's Dark Moon Rising & Trinity, or the overrated but absolute must-reads The Killing Joke & Arkham Asylum: A Serious House on a Serious Earth, written by legends Alan Moore ( Watchmen , League of Extraordinary Gentlemen , Swamp Thing ) and Grant Morrison ( Batman RIP , Final Crisis , Animal Man ), respectively.
From there you can either read through important stories collected from the ongoings (monthly comic books) or cheat yourself out of good reading and look on Wikipedia. Either way, your ultimate goal should be to have a decent enough grasp on Batman lore in comics before taking on Grant Morrison's "six-years-and-counting" epic Batman masterpiece, beginning with 2006's Batman & Son (which has one of the greatest openings ever seen or read) I will warn you though, there has recently been a reboot at DC that retcons certain aspects of character's histories and sometimes entire stories (expect to hear of this more often than usual if you're looking to get into comics) Thankfully though DC have handled this one much better this time around, and if all isn't explained in the first issue/story, then the specially-made Zero issues will fulfill this purpose.
Anyway, I wish I could wholeheartedly recommend some non-superhero comics, but I ain't the type of guy who does that unless I've actually read it myself. Of the small collection I've built though, the titles Nightmare Factory and God Save the Queen come to mind. Like I said though, I still haven't read them. I'll be sure to link you to some guddens when I manage to get round to it.