Are you interested in comics in general, or superheroes in specific? Because I gotta be honest - as a long-standing fan of comics, it's worth knowing that 90% of superhero comics are absolutely not worth reading if you're after something beyond "Male Power Fantasy In Spandex Punches Stuff Until It Dies".
Comic superhero stories worth reading tend to be those that either deconstruct the concept, or else are written by what I suppose you coud call "comic auteurs"; that is, writers who specialise in comics as a storytelling medium, rather than novels, and who run their own creator-owned comics, rather than simply doing the same old intellectual properties that Marvel/DC/Whoever have been churning out for years.
If you're interested in just disposable superhero fun, there's honestly not a lot I've read that didn't make me want to give up the medium entirely. "The Twelve" by J. Michael Strazinsky is pretty good. "Iron man: Extremis" is very enjoyable. Any "Punisher" collection by Warren Ellis is excellent (though not for the faint of heart; the series "Born" features some absolute grotesquery, and "The Slavers" may be the most horrible story Marvel ever published - basically the author had read about human traffickers and got so angry that he had the Punisher kill them. It's well worth reading, but will leave you upset unless you're dead inside).
People worth reading:
Garth Ennis. He's a Belfast-bron writer who started out on Judge Dredd, before writing for DC. His comics are characterised by black comedy, foul jokes, absolute misanthropy, pure hatred of Christianity, and a general loathing for superheroes. On a good day, his work is excellent - his above meontioned run on "Punisher" will probably never be equalled. On a bad day, his work is childish and the jokes fall flat - I don;t particularly like "The Boys", because it's, well, lame. I know a lot of people like it, but it's just his 2000AD stuff redone for an American audience, so there's very little decent characterisation in amongst all the scatological "humour". His creator owned series "Preacher" is well worth reading for the characters, though it is frequently deeply unpleasant. Don't read if you're a Christian, as the Christian God is explicitly the villain of the piece.
Grant Morrison. His heartfelt, deeply humane run on "Superman" is the only time I've liked the character. Similarly, "Arkham Asylum: A Serious House on Serious Earth" is the only Batman story worthy of mention (including Alan Moore's "The Killing Joke", which is interesting, but all of thirty pages, so not really a read). His creator owned series "The Invisibles" and "The Filth" are amongst the best books I've ever read ("The Filth" especially), but be warned: they are very, very weird. Like most of his stuff. His impossible-to-find comic "Kill Your Boyfriend" is glorious fun. "We3" combines his love of animal rights and cyborgs and is brilliant. His run on "New X-Men" is also well worth anyone's time - it's a massive deconstruction of the whole X-Men mythos (where he points out that a series ostensibly about change never does).
Neil Gaiman. Just, seriously, read "Sandman". All of it. All ten books. It is the best thing I have ever read. I have yet to find anything remotely close to how good it is. I doubt I will.
Joss Whedon: His run on "Astonishing X-Men" is the best the comic has ever been. His work on "Runaways" is also brilliant.
Bryan K. Vaughan: "Runaways" is unashamedly my favourite pure superhero comic ever. His creator-owned series "Y: The Last Man" is also very good.
Warren Ellis. Very, very hit or miss. The stuff he wrote for Marvel during what he called his "Year of Whoredom" is dreadful (with the exception of the aforementioned "Iron Man: Extremis", and I suppose "NEXTwave", if you like that sort of thing. Which I don't). However, his creator-owned work is very strong. If you're after superhero stories of his, I recommend his "Fear The Superman" trilogy ("Black Summer", "No Hero", and "Supergod", of which I would say "Black Summer" is the best), as well as the "Planetary" series he did for Image. Other people will always recommend "Transmetropolitan" (Hunter S. Thompson fight Tony Blair IN THE FUTURE!!!) but I found it fairly weak, especially the issues after Ellis' father died (where he's obviously so cut up inside he couldn't be bothered with fiction, and frankly, that's fair enough. It does mean "Transmet" has a much stronger start than finish, though). I also love "Global Frequency". His run on "The Authority" isn't bad either, largely because he writes the titular groups of superheroes as the supervillains they actually are.
Gail Simone. Her work "Secret Six" is very hard to get hold of, but very good. She writes Bane as a badass, which is something he sorely needed.
Alan Moore. Don't bother with "Watchmen". Everyone'll recommend it, and it's kind of obvious, as well as very much a product of the 80's. Read "Top Ten" instead - there's far more female characters, and it's far more interesting. I guarantee you'll never see its central idea anywhere else ("What if there was a city where everyone was a superhero, from the hotdog sellers to the homeless?") Nice to read about some gay characters who aren't gay to get a quick headline and free press too.
NAMES TO AVOID:
Frank Miller. Did one good comic ("The Dark Knight Returns"). Spent the rest of his life attempting to destroy any credibility he had built up. Finally succeeded with "All Star Batman And Robin". Frank Miller is a lunatic, his comics are a joke that's been old for a decade. Give his stuff a go if you like, but be prepared to regret your wasted time.
Mark Millar. There are no words for how much I detest this guy. He's basically a 13 year old boy who's never done anything meaningful in his life, but boy oh boy has he watched a lot of movies! His work is uniformly misogynist, misanthropic, joyless, and cruel, but with a patina of style that people seem to embrace. Give his work a go - the "best" is probably "Wanted", which has some admittedly clever ideas, but altogether too much casual rape for my liking.