Bradbury remains the only writer whose works I've read and enjoyed more than twice. It's a shame that many people encounter him only as kids; I know when I first read Fahrenheit 451 I was twelve years old, and as far as I was concerned it was a slightly odd story of a guy who set stuff on fire for a living, which ended with a scary robo-dog and a nuclear war. Two readings since then have expanded my appreciation of it. The richness that bears repeated readings in any work is found in the shadings of human experience, and these suffuse his work, like myths which constantly yield new insights in each new age. In fact a character in Fahrenheit 451 points out this very richness as one of the characteristics of true literature. As I was re-reading some of his stories in the past few months I was struck by the following themes.
1) Human beings can't help but bend the world around us into a reflection of ourselves. This is tempting, but ultimately smothering and toxic. This occurs most obviously in "Mars is Heaven" and "Here There Be Tygers", but most of the stories in the Chronicles features some aspect of people hurting or killing themselves through this comforting blanket of self-deception. I'm always struck by how the telepathic Martians who facilitate the deception seem to need to help us do this, and need us to need their deception, though I'm not quite sure what to make of it. (The Martians are surely in part a symbol of Native Americans in our history - something onto which we project our fantasies of self and other - and Bradbury even tells us it was our diseases that wiped them out - but that's not all they are.) The most prominent demonstration of the Martians' need for us can be found in "Mars is Heaven", where Martians kill the Earthmen to prevent them from piercing the veil of the dream they'd all allowed the Martians to erect around them. Still, even after the Earthmen are dead and there is no use to further deception, the Martians appear to mourn them, or at least the comforting illusions the Earthmen brought with them and that was now slipping away. The Martians just can't help but receive and amplify the fantasies and memories of the humans.
2) There is a subtle deathwish theme that runs through Bradbury's stories, expressed most nakedly in "The Blue Bottle", but also present in "The Earthmen".
3) Bradbury was able to avoid the more distasteful mid-20th century American literary themes, the badly-aged ones of decrying commercialism through chains of made-up Brandnames(tm), or baldly Freudian character explorations. As for the former, he does certainly register his frustration at 1950s America's Detoquevillean conformity (Fahrenheit 451, "There Will Come Soft Rains", "The Earthmen"). As for the latter, the excellent "The Veldt" may be the closest he comes (after all, there's a psychiatrist in the story) but many of his stories have characters with a moral and psychological simplicity that, if you pay attention, you'll notice being compared unfavorably to one or more other characters (most of his astronauts, or the country folk in "The Burning Man".)
4) It's difficult writing about his stories because Bradbury doesn't place hard little gems inside his stories - you can't point to the page and say "this is my favorite part" - but rather you slurp it down and enjoy it afterward like a warm meal on a cold day. "Kaleidoscope" and "There Will Come Soft Rains" are especially poignant examples. Interestingly I've never once heard a hard scifi geek (and I proudly count myself one) complain that Bradbury's science is unrealistic. You don't stick around long reading Bradbury if you want to know about Martian biochemistry or how his rockets work. But his writing is most certainly speculative fiction, in that Bradbury is playing with the element of setting so that he can tell a story. It is not science fiction, in that he clearly doesn't care about the real atmosphere of Mars, or if there are jungles on Venus, or anything else other than our imaginings of these things. And neither do we, when we start reading the stories he put in these places. Possibly related to this, Larry Niven once made a comment to the effect that young writers would break their hearts trying to imitate Bradbury, because his stories somehow just happen without any respect for action or structure.