If you're getting into comics now, it works out for you pretty well. DC is having a relaunch of their entire line, but canceling all of their old books & restarting with 52 'new' titles. Some are hit & miss, and they don't feel like true relaunches, more like slight reworks. Currently the books are all the rage & speculators have been swooping in to grab as many books to resell on Ebay. So you may want to look into digital copies, as DC has gone into same date releases as their physical copies.
Books you want to take a look at are Action Comics by Grant Morrison who is doing a sort of earlier version of Superman, so he's wearing his S shirt & cape, but with jeans & shoes. It's sort of the forming of the ideology of what becomes the Superman we sort of know. Animal Man, a very trippy book. Batwoman who is a mystical of sorts oriented Bat character, she's also known for being a lesbian main character. Not a focus, but a note about the character. Green Lantern who became very popular in the last few years, the book has an interesting twist if you know it's basic history. That's just a few.
Marvel is a bit harder to recommend because they are in the midst of their annual big marketed crossover event 'Fear Itself', and it's really not that interesting. Also X-Men has started a big event, as well as Spider-Man, so you'd be jumping into the middle of things with many Marvel Books.
So you may want to consider some alternatives. Atomic Robo is a fun science, pulpy type of book. Invincible is a more true to form super hero book than anything DC or Marvel is doing these days. There's of course Walking Dead that inspired the popular TV series. It's hard to recommend titles without knowing what genre you're interested in.
I'd recommend checking out a public library as many carry large collections of graphic novels. So for free you can check out entire story lines of popular Marvel & DC characters, Japanese Manga, and some of the more revered alternative titles that you probably ( Maus for instance ) may not have heard of.
Hope that helps.
Me personally, I tend to get books featuring art I like, over story lines. I do read Invincible & Atomic Robo regularly, enjoyed one or two DC titles, and just get the random Marvel.