Below is a walkthrough for the Raise Weapon Skills task in Rune Factory 5.
Client: Scarlett
Objective: Increase one of your weapon skills to level 15
Reward(s): Rush Attack x1
After accepting this task from the Task Board, Scarlett will ask you to level one of your weapon skills to level 15. At this rate, there is a good chance you could have one weapon skill to 15 in which case you can turn in the task at the Task Board. Otherwise, pick a weapon then head out in the fray and keep defeating enemies until the weapons level reaches 15 then return to the Task Board.