If there's any reason flashbacks have become faux pas in modern writing it's because of the Saw series. Untangling the web of twist accomplices is the key to unlocking the chronological order of the Saw franchise. The only problem is the origin story is split across various movies.
The only proper way for it to all make sense is to order the entries by the flashbacks first, with the list of movies they occur in, then actually listing out the rest of the movies in order. The Saw movies are always good bloody fun, but there is so much story behind the surface gore of it. Here's the chronological order of the story in the Saw movie series.
10 Backstory 1 (Saw IV & Saw VI)
It all starts with Saw IV where we meet John Kramer, an engineer, and his wife Jill. Jill was pregnant but opened up a drug clinic in which an addict by the name of Cecil Adams tried to rob for drugs for another addict and soon to be an important character, Amanda, as revealed in Saw VI.
Cecil slams the door against the pregnant belly of Jill, causing a miscarriage. John drifted away completely traumatized from the event and eventually broke up with Jill. He isolated himself with his thoughts until he was diagnosed with cancer.
9 Backstory 2 (Jigsaw & Saw IV)
Being embittered by his wasted life, the future killer watched others closely and observed how they lived their lives. Seeing others waste their lives while his life was coming to an end depressed John. After being denied money for his cancer treatment by William Easton, in Saw II, John survives his suicide attempt by driving his car off a cliff, leading to an epiphany that spawns the development of his ideals.
Back to the Saw IV flashbacks, John designs his first traps for Cecil. In Jigsaw, Logan implies that he survived one of the earlier tests and became John's first apprentice.
8 Backstory 3 (Saw IV, Saw V, Saw II, & Saw VI)
In Saw IV's flashbacks, John adopts the idea of "Instant Rehabilitation one person at a time" and continues on his killing ways using traps designed by Logan. He's eventually labeled the Jigsaw Killer by the media when his signature is found through removed jigsaw puzzle-like pieces of flesh from the victims.
Jigsaw's controversy eventually leads, as revealed in Saw V, to a copycat killer in the form of a cop named Mark Hoffman who would become a very important character. Having kidnapped and blackmailed Mark, Jigsaw annexes him to his cause. Mark eventually designs traps to be used in the first movie, where the timeline does not get simpler. Prior to the first Saw movie, Amanda Young becomes the first known survivor of Jigsaw's games. As revealed in Saw II, she takes to him like a father figure and says his games benefitted her life, becoming another apprentice before the first movie even starts. As shown in Saw VI, Jigsaw then shows the "rehabilitated" Amanda to Jill, who essentially goes on to become another accomplice/apprentice.
Saw technically starts with Amanda kidnapping Adam Stanheight for the game. Jigsaw, however, also goes for kidnapping his doctor who diagnosed him with cancer, Lawrence Gordon. This movie introduces Jigsaw's ideas and crazy blood-soaked gore games.
An important moment that leaves a lasting impact on future entries is Jigsaw helping the doctor and making him another apprentice. Folks, we are only past the first movie and there's already 6 different Jigsaws.
6 Saw II
Another game is underway, with Amanda serving as a red herring. Eric Matthews is introduced as another victim who in the end gets kidnapped by Hoffman for a future game, as revealed in a flashback from Saw IV. Besides that, there isn't much that happens that is relevant to the overarching plot of the series.
The movie's point seemed to be about establishing that Amanda was actually evil, even if the story revolved around her getting a reckoning against Eric Mathews for wrongly accusing and jailing her for a crime.
5 Saw III/Saw IV
These two movies happen at the exact same time timeline-wise and follow two of John's protegees, Amanda's traps for Saw III and Mark Hoffman's traps for Saw IV. Amanda's starts off as a disaster because she no longer believes Jigsaw's trap actually change people, even though she was one of them, and starts making inescapable traps that just kill people. Jigsaw eventually has to put her through a test so that a straight-up immoral murderer isn't in charge of his legacy, which she fails and gets them both killed.
Saw IV follows Mark Hoffman as he brings Eric back in for a test with David Riggs, a reoccurring SWAT member investigating Jigsaw. Mark eventually finds the bodies of Amanda and the original Jigsaw. The autopsy scene is actually the last scene in the canonical timeline for this film, culminating with Hoffman finding a tape inside Jigsaw's body telling him that he will be tested still.
4 Saw V
In Saw V, Jill is given a tape by the police from her dead serial killer husband with a list of instructions she must do.
The movie itself follows the new psycho Jigsaw in another game much like a child's first ride on a bicycle. Hoffman escapes police pursuit to set up the conflict in the next movie.
3 Saw VI
Following instruction meant for Jill from Saw V, Mark goes after the man who denied John his cancer money, William Easton. Mark lets Jill know he's taking over the position as Jigsaw, which does not ring well with her as she has instructions to kill him from John.
Fittingly this was probably the test John forwarned Hoffman about. William meets a horrible end and Mark goes back to the scene only to get jumped by Jill who puts him in a reverse bear trap, much like the one he designed for Amanda in the first Saw. Mark eventually escapes, even though mangled.
2 Saw 3D
Mark Hoffman and Jill Tuck are battling for the title of Jigsaw. The storyline in the Saw movies is finally starting to simple out. Another person is kidnapped for brutal games, while Hoffman goes on the attack to find Jill, who is in police custody outs.
Jill outs Mark as Jigsaw which he counters by mowing through a police station to get to her and end her life in a Reverse Beartrap. Another apprentice Dr. Lawrence Gordon, from all the way back from the original Saw, reveals himself as a new apprentice and finds Hoffman, finally putting an end to him.
Jigsaw, the movie not the killer, can be watched without prior knowledge of all the other movies, but still adds to the story slightly. It takes place 10 years after Saw III, with another apprentice rising after Jigsaw's death.
Much like Hoffman, it's much the same where everyone dies and he gets away. With no follow up to this in 2 years and a reboot coming out made by Chris Rock and Samuel Jackson, it's pretty easy to say you don't have to really see this filler movie.