The puzzle missions are accessed via the Main Menu by selecting Single Player, Puzzles. The boot camp tutorial, campaign and puzzles are all possible to complete immediately after the introductory tutorial due to a cheat code Team 17 (the developers) added to the game as an apology for some of the bugs that were included in the original release. To activate the cheat code, from the Main Menu select Help & Options, scroll down to credits, then press , , . At this point you will need to complete any one level to unlock "Fast Learner", "The Trooper", "One for the Money", "Upstairs Wares", and "Worm Winner". If you have the Mars Pack DLC, you will also unlock "Strife on Mars".
The quickest level to complete is part of the boot camp tutorial. From the Main Menu select Campaign and the selector will default to a rat icon labeled Sewer at the bottom of the screen. Press and you will see the description for Training 1: Weapons. Press again and complete the quick tutorial level to unlock the achievement.