Note: You can also unlock a trophy from upgrading a weapon for the first time. Generally speaking, the Stability upgrade will be the first thing to prioritize on guns like the rifle, pistol, and bow. However, if you are an experienced player, upgrading the Stability of the Bolt-Action Rifle will help you increase the accuracy of your shots giving you a better chance of conserving your ammo when landing headshots and efficient kills. In addition, weapon upgrades like extra capacity for all guns will be useful, as you may have trouble making each shot count, so that extra bullet capacity will give you extra chances. This allows you to one-shot most enemies just by shooting their chest area which is extremely useful especially If you are having trouble aiming at the enemy's head.
If you are a beginner at the game, it is recommended that you focus on upgrading the Damage of your revolver. Which Weapon Upgrades Should You Get First? For Beginners