The first books I read were the Millennium Trilogy, by Stieg Larsson.
Starting obviously with book 1 - The Girl with the Dragon Tattoo:
Disgraced journalist Mikael Blomqvist is hired by Henrik Vanger to investigate the disappearance of Vanger’s great-niece Harriet. Henrik suspects that someone in his family, the powerful Vanger clan, murdered Harriet over forty years ago.
Starting his investigation, Mikael realizes that Harriet’s disappearance is not a single event, but rather linked to series of gruesome murders in the past. He now crosses paths with Lisbeth Salander, a young computer hacker, an asocial punk and most importantly, a young woman driven by her vindictiveness.
Together they form an unlikely couple as they dive deeper into the violent past of the secretive Vanger family. (from the Steig Larsson website)
This first book is relatively self-contained, you could easily stop after reading it. However, if you are hooked like I was, you will quickly want to move on to book 2 (The Girl who Played with Fire) & 3 (The Girl who Kicked the Hornets' Nest). However, a warning - if you read #2, you will want to immediately pick up #3 to see what happens, so make sure you have enough time to read both!
All three are gripping mystery/thrillers, with clever and original story lines that draw you right in. All three have been very well-received by reviewers and I agree with them - they are great reading. They do have some very unpleasant and violent scenes, but more often the violence is hinted at, rather than actually expressed.