I started by attempting some of the recipes in the Moose::Cookbook with a few personal variations and then moved to Moose::Manual since I am more of a learn by doing type. Attempting a few of the recipes with my variations first to get a flavor of Moose was the fastest way for me to get into the Moose world.
Moose isn't just a cool object oriented framework, it also encourages (not requires) a certain object oriented style. Getting a handle on the style early on will allow you to make the intuitive leaps needed in learning to reach that point where you are using Moose and can extrapolate expected Moose behavior that you may not have learned yet. The ability to intuit the possible way Moose would solve a problem takes you a long way to find the answer for solving your problems. So then you will be able to identify when you have reach a point in your code where what you are attempting to do in Moose seems unnecessarily complicated. In most cases this means you have reached a point where it's time to go back to the documentation.
Using this method allowed me to avoid trying to digest the whole Moose cookbook or manual at once. (Don't try to eat the whole Moose in one sitting.)