Different characters have different primary weapons. Harris is the only character to have a full auto weapon (a M1917 Browning). Spending Blood Shards on your Weapons in the weapon menu can upgrade your primary weapon's stats such as reload speed, magazine capacity, etc, while slotting Blood Shards in the Elemental slot can fundamentally alter the way the primary weapon behaves. Also, successfully hitting the Active Reload target when reloading a weapon will provide a damage boost to your next clip.
Some characters have slower firing weapons, so don't be stingy about using their Devil Powers to compensate.
Cheers!
Kyle