Adventure Mode was originally an option on the main menu but now it is found simply by playing the game. In each game world you can find one instance of Maxwell's Door which can be opened to try Adventure Mode. This door can be stumbled upon but it is can also by found by using a Divining Rod. It is the only other object aside from the Five Things that is found by the Divining Rod.
Once you find the door and enter, you'll begin a five chapter (plus epilogue) quest to catch Maxwell. You'll always start at chapter 1 no matter how far you've gotten before and each chapter will test you to your limits. There are as many as six different types of Worlds that can appear. For the most part, the first four chapters can be any combination of worlds at random while the fifth chapter is always Darkness. No World will appear more than once in a playthrough.
When you start, you can't bring any items or unlocked prototypes with you. If Wilson has grown a beard, it will disappear. Whenever you enter a world, you'll always start near a Divining Rod and sometimes near a stockpile of useful supplies for the world you entered. From there, you need to survive, find the Five Things and get to the next world.
Upon leaving for the new world, you'll only be able to bring four items with you (though stacks only count as one item) and any prototyping knowledge from previous worlds. You should try to take a Walking Cane with you if possible as well as a particularly good food supply such as Honey or Jerky. After that, you should try to bring items useful in future worlds. You may not know what is coming but try to guess based on what has come before. Like the normal mode of the game, any buildings pre-built but not placed will come with you and be placeable in the next map.
The following pages will cover the different worlds you can encounter but succeeding will still not be easy especially since the levels are randomly generated and can happen in any order.
By completing Adventure Mode, you'll unlock both Wes and Maxwell as playable characters.