We could write a whole book about Project Mayhem, but it's already been done. Yeah, it's called Fight Club.
From the beginning to the end, even when our narrator isn't talking about it or thinking about it, Project Mayhem is there. Not to sound creepy, but your waiter, the person that does your dry cleaning, the clerk at the convenience store—they could all be members of Project Mayhem. As they say, "It's useless to fight us" (26.56). Gulp.
Project Mayhem's game plan is to "break up civilization so [they] can make something better out of the world" (16.76). Pretty self-explanatory, we guess, and they attempt to achieve their goals through both subtly subversive and in-your-face explosive means.
But here's the thing. Tyler can't do it alone, so he has to recruit people. And how does he do this? Through fight club, exploiting the men who join—men who are literally fighting for their lives. Tyler's charisma gets them on board though. Just listen to the guy:
"The trigger [...] frees the hammer, and the hammer strikes the powder." (16.45)
These young men who feel they aren't living up to their potential are kegs of powder, and Tyler is the trigger. The impressionable young men are just tools for Tyler to achieve his goals.
Given that Tyler's entire existence managed to subvert our narrator's consciousness, it should come as no surprise that subversion is their main method of attack early on. The seeds of Project Mayhem were growing in our narrator's mind even before he realized it. One of his haikus actually captures Project Mayhem's philosophy in seventeen syllables:
Worker bees can leave
Even drones can fly away
The queen is their slave. (8.2)
Project Mayhem's methods are funny at first, like the bumper stickers they paste on unwitting cars: "I Drive Better When I'm Drunk" (18.102), "Make Mine Veal" (18.102), "Drunk Drivers Against Mothers" (18.103) and "Recycle All the Animals" (18.104). Those are the work of the Mischief Committee. But eventually, mere mischief isn't good enough. Once Tyler is found out, the funny subliminal stuff doesn't work anymore. At that point, they have to kick it into high gear, killing government officials and blowing up high rises.
All the violence and death caused by Project Mayhem points toward one goal: "A near-life experience" (19.3). It seems counter-intuitive, but it all ties into one of the book's main philosophies that you can only live once you hit rock bottom. And as Project Mayhem shows through its progression from mischief makers into full-blown domestic terrorism, getting there doesn't happen overnight.