n my opinion, this is one of the most important masteries to advance in as it is required to be able to upgrade all armors above +10. for example, to get your armor to +11 you need a behemoth mastery of 25. Other benefits are cores, more potions, double break parts, added elemental resists, etc.
The fastest way to level this mastery is to complete the first 4 objectives for every behemoth in the game. This means that you need to craft every armor piece in the game, upgrade them to +6 and preferably to +10 when you can, and use the armor of the same type as the behemoth you are fighting 10 times.
For example, if you are up against a Drask, you need to wear Drask armor to complete the objective. It is important to note that always you need to wear a full set of the specific armor to complete that objective.
You should also attempt to break parts of every behemoth you hunt. The first tier of this objective is to break 25 parts, which is easily doable.
After this, it becomes harder to advance in the behemoth mastery. Just continue to focus on completing the objectives and try to be patient.