I adore DC and Marvel comics, and find the former much easier to recommend thanks to the "key" stories being easy to get hold of and enjoy without going through anthologies. My passage into them was very unusual though.
As a teenager I thought they were pretty dumb, and I liked to think that manga was a much more mature form of comic for mature people such as myself. However, a friend lent me the Killing Joke around the time Batman Begins was being trailed and I was absolutely stunned by it. From there I took a leap into Watchmen and The Sandman, which my friend lent to me. Watchmen cemented my interest in Alan Moore and Sandman gave me a little wedge into the DC Vertigo stuff.
My friend lent me Saga of the Swamp Thing and, while it was a pretty silly concept (man becomes a plant monster that can control vegetation), I was hooked on it, and pretty much bought all six volumes in the space of a few months. From there it was easy to start reading more of the characters I was interested in, I read Hellblazer and most of the "key" Batman stories. I discovered writers and artists which helped me follow into Marvel with books like 1602 and Daredevil: Born Again (Gaiman and Miller respectively). From then on my collection grew rapidly over the space of eleven years (I'm 28 now, most of this happened back in 2006 when I was seventeen and edgy). I have loads of volumes (not issues, I just want to read the stories rather than collect things) from DC, Marvel, Dark Horse, Image etc - but DC and Vertigo takes up most of my shelf space.
If anyone wants to get into them I'd recommend checking out a character that catches your interest. Unlike Manga or many indies, long running characters are more like concepts than canon stories. So you'll find loads of different interpretations of them over the years. You don't need to read ALL of Superman to enjoy the wonderful All Star Superman. You don't need to know anything about Daredevil to enjoy Mark Waid's run on the title. Most writers do a soft reboot of the characters when they take over, changing the direction and tone to reflect their own style, as such you can find great "runs" that don't need much work to untangle.
Edit - I just realised I haven't given a very good opinion on them. I enjoy them because they kind of feel like our society's mythology. When you're reading a great Batman story or watching a lesser known character get a perfect reinvention it's like reading something that resonates with popular culture at large. These characters have inspired billions of people and have even changed the world in no small ways. That's really cool to be a part of.