A lot of people who read Gibson bash his writing because of the superficial nature of his characters. They're missing the point. One of the overarching themes of cyberpunk is the idea that technology, while revolutionizing our world, is also slowly stripping us of many of the things that make us human. Cyberpunk is a dystopia, not a Star Trek style utopia. The people living in the Sprawl, in Gibson's world, are superficial, because that's all that they have. It's professionalism taken to an unhealthy degree, and it mirrors many people that I know. Not everyone in the world has a deep, complex personality. I dare say a significant number of them don't.
That being said, Gibson truly shines as a short story writer. As a fan and practicioner of the short story, the sparseness of plot and space in general is one of the strong points in cyberpunk. The genre is about impact and style, and the shorter the story, the stronger the impact has to be to justify the experience.
Burning Chrome is a book with ten such impacts. The first story, Johnny Mnemonic, probably has the best chance to be known by the general public, and has the hardest reputation to overcome. It's great scifi, albeit far from the best story in the book. As other reviewers have said, it does introduce Molly and gives a great taste of the Sprawl. The killing floor is also one of the best examples of culture in Gibson's world.
The Gernsback Continuum is one of those stories that border on the incredibly bizzare. I didn't care for this story as much as the other (it's probably my least favorite), but ti's still bizzare enough to be entertaining.
Fragments of a Hologram Rose if I remember correctly is one of Gibson's first short stories, and it sets the stage beautifully for his style of writing. Short, sweet, and encapturing a moment in life, it may not have the depth of impact of the sprawl stories, but it's still high quality. Gibson deals with emotion, and the avoiding thereof in a painfully human manner. Namely, the characters avoid it at all costs.
The Belonging Kind is another story, sci fi in it's premise, but not so much in it's execution. Refreshing in how alien (litterally) socially popular people can be.
Hinterlands is my second favorite story in the book. It's what I call "classic" scifi- it's set in space, on a space station, and involves man's exploration of the unknown. But instead of a star trek slant, it plumbs into the desperation of wanting to know, the craving from the tree of knowledge for more. I identified with the desire to Know that which is unknown, even at the cost of your own life. A very strong story.
Red Star, Winter Orbit is another "classic" scifi tale, about a decaying russian space station at the end of an era of war. Probably ranks in my 3 least favorite of the book, but that's like saying that cake is a little dry.
New Rose Hotel is another personal favorite, about corporate espionage and the art of the double cross. This is linked to the sprawl stories, although it's hard to tell (certain companies are involved, and their outcome seems to be reflected and alluded to in the sprawl trilogy). Powerfully narrorated by a man about to die, it's light on the sci fi, but strong on Gibson's style.
The Winter Market is an eerie tale about desire and raw drive. The interesting thing about this story is that a few people, including myself, have written stories before ever reading Gibson that share many of the same ideas and themes of this story.
Dogfight is another one that's different. A story about desires, values, and passions, between a convict with a mental block and the college girl he meets with an equally strong mental block. Dogfight refers to a holographic game that the main character is obsessed with.
Burning Chrome. We finally get to the namesake of this book, and we find Gibson in full stride, in lyrical command of his genre. Like Molly from Neuromancer, everything is right in place for Burning Chrome. His moves are down pat, his style oozes, and the delivery of the mood and atmosphere hit you like a jackhammer. Basic plot? Boy meets girl, boy goes on hacking job for girl so he can retire, girl falls for boy's chum and assistant, girl splits as boy & assistant strike it rich. Simple plot, but oh-so-powerful in it's delivery.
Overall, Burning Chrome is worth the investment. Don't read Gibson to have intriguing, incredibly developed plot and characters, read it for the mood and style of the writing. Each page is like a work of art, carefully crafted to leave an emotion, an impression. In a fictional world where life is fragile, the easiest way to achieve immortality is to leave a lasting impression. This is the goal of all the arcane grammar, the lyrical prose, the point-blank blast of imagry that Gibson throws up. Don't try to analyze the plot, analyze the mood. It drips in each of the Sprawl stories.