Unlock all shop items.
Find all spare parts.
Theres a map here
Glitch here (be sure you have around 32 items before trying the glitch or it may not work. I grab a bunch around the west cost, flew around with a Helicopter where they all were, and once I flew over them all, I saved and reloaded. This will allow you to re grab them all. You can get around 20 each time you do this, without find any new ones. Check the link for more info.
Beat the story (you unlock the nuclear bunker buster!)
Capture all outposts for each faction.
HVT missions (See justice for all)
Destroy structure missions
Destroy 40 billboards (once you have the PLAV as a faction)
Kill 50 UP
Kill 50 PLAV
Kill 50 AN
Kill 50 Chinese
Warning, if you beat the game, you may not be able to do certain missions for the faction that left Venezuela. So make sure you complete all the HVT missions, outpost missions, and faction missions before you beat the game to be safe. Be sure did everything before you complete the last two story missions, so you won't have to re-beat the game.
If you are having trouble with the challenges do them in co-op. There extreme easy with a friend. You can also check how many items that are unlocked in the progress menu by pushing the back button.
Thanks to MrIssues and MetalReaper for the other ways to unlock the shop items.