Three books by Stieg Larsson, five more by two other authors, and three screen versions that share nothing but the source.
The answer splits depending on whether you want to read or watch, and the reading answer is the simpler one. Start with the first novel. It stands alone, so if you bounce off it the series is not for you, and if you like it, books two and three are effectively one book split in half.
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A self-contained mystery and the strongest way in. Note that the English title was chosen by the UK publisher against Larsson's stated wishes; his own title translates as Men Who Hate Women.
Treat these as one book. The second ends without resolution and the third finishes it. Stopping after book three is the complete, intended ending of Larsson's story, and most readers should.
The strongest made version, in English, telling book one completely, with no sequel you are obliged to chase. Its lack of a cliffhanger is an accident of the sequels being cancelled, and it works anyway.
The Swedish trilogy is the only screen version that covers books two and three at all, and this six part cut is the same cast and the same footage with roughly 110 minutes restored. The gain is largest on films two and three, which were compressed hardest for cinemas. Around nine hours in total.
Then what? Books four to six by David Lagercrantz are competent authorised continuations by a different writer with a different voice, and books seven onward by Karin Smirnoff shift again, more literary and less procedural. Take them only if you want more after Larsson's ending. Do not start at book four just because the 2018 film did. And do not start with The Girl in the Spider's Web: it adapts book four, skips two books, recasts everybody, and its commercial failure ended the English film line.
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