So I've been searching for certain arts for my class braver(bow). I can't find the starting ones in player shop. Are they completely a random drop, or is there a way to specifically find them.

When you select a mission you can look for an option that says "quest log" under quest log you can check for enemies and loot dropped in the quest. I cant give you the exact directions atm because I am reinstalling the game for the bagillionth time but here is a link to a youtube video that can help you: ---> Here ... I hope this helps