Of the three, The Girl With the Dragon Tattoo has the tightest story and the best characterization, I feel. Also, most appraisals of the subsequent books are bound to be unfair, because Larsson apparently died after having turned in the three manuscripts, so he wasn't around to edit and polish with publisher input. Or maybe he edited and polished with private publisher input; I don't really know. But I feel that the second and third book needed some overall work, and it couldn't all be blamed on being translated from Swedish. I've read translated novels before without such issues (Let the Right One In by John Ajvide Lindqvist was also Swedish, as an example). Apparently, the series was also intended to be ten books long; faults aside, I regret that did not come to fruition.
So: The Girl With the Dragon Tattoo. We're introduced to the main players: Mikael Blomkvist and Lisbeth Salander. He's a journalist who takes journalism to its Platonic ideal of championing the Truth and Justice and suchlike. She's a social misfit who's a brilliant hacker. Blomkvist first, but then Salander, are hired to solve a rather old "locked room" mystery, wherein the niece of their boss disappeared from an island thirty years prior, on a day when the island was both busy and cut off from the mainland. There are other characters that appear throughout the novels, but these two are the "most important", and frankly, the ones I felt the more compelled to follow. Arguably, Erica Berger (Mikael's Millennium partner and sometimes bedmate) is important, but less so until the third book.
The Girl Who Played With Fire: Blomkvist discovers the murdered bodies of two researchers who were working on a rather exposing project for and with him, and Salander is, improbably, the prime suspect. This book frustrated me. The bulk of it is a police procedural in which, due to the omniscient style of narration, we the readers are watching the police do it wrong. And see where the deliberate monkey wrenches are thrown into the investigation. And wonder where the heck it is Salander disappeared to, because wasn't she supposed to be in this book too? There's some great government coverup stuff in it, but overall, I was pretty unsatisfied. But, Larsson wisely ended the book with a bang, and so I needed to read the next.
The Girl Who Kicked the Hornet's Nest: The book picks up pretty much immediately after the prior one, and by immediately, I mean within hours, anyway, not the next sentence. This book had a lot less to wade through before the story was reached, I felt, and overall, it was also a tighter and more complete book than The Girl Who Played With Fire.
Overall, I don't regret the time I spent reading the books. I've also watched the first two Swedish movies at this point, with the third in my near future. Given the levels of violence, sexual and otherwise, in the books and the movies, I'm a little surprised at the widespread audience appeal. I mean, my grandmother read these. And watched the first movie. I hadn't yet read The Girl With the Dragon Tattoo when she mentioned that she'd read them, so I didn't know. But now that I do, it's kind of funny to me, in the way all of the older set were reading A Million Little Pieces a few years back; I know it was supposedly a tale of redemption and all that, but it was also a tale of explicit and frequent drug use, among other things.