In the weapon selection screen under "Materia & Equipment" in the main menu, you will find the requirements for mastering a weapon's proficiency in "Ability Info" under "Proficiency Bonus" when hovering over the selection. Every weapon will have different requirements, and players should complete these requirements as they progress through the story, making sure to slot characters with max weapon proficiency into the active party. Maxing a weapon's proficiency allows you to use that weapon's ability on another weapon. Completing these as soon as possible will save you from having to grind them out later for  Well-Rounded.
 

As an example, in the above image, the requirement for increasing the Buster Sword's proficiency is to stagger an enemy. Repeatedly meeting this objective will raise the weapon's proficiency, which can be tracked by the blue progress bar under the weapon's ability. Alternatively, you can increase the weapon's proficiency by using its respective ability, but it will increase more slowly if you do not meet the objective. You will receive a notification in combat when the weapon's proficiency has been maxed out. You can also identify this by the checkmark that will appear next to the weapon's name.