The Surge 2 has a straight-forward weapon upgrade system. You have to kill enemies, collect enough parts and then the Upgrade section to increase the weapon capacity. Everything will be done through the Medbay, even crafting new Gears. To upgrade weapon hit the nearest Medbay and follow the instructions below on how to find raw materials for weapon upgrades.
How To Upgrade Weapons
After having a weapon like the Salvage Gauger, take down a few enemies. There are two types you will face during the start of the game, an enemy with exo-suit and another prisoner type who carry weapons. When you kill them they will drop schematics to craft new gears and important raw materials in the form of MK. I or MK. II, etc. These are the necessary drops you will need to upgrade a Mavweapon.
After having some of them go to Medbay and Navigate to the Upgrade section. You will see a small settings icon on the top right that indicates the weapon is ready for an upgrade. There can be 10 to 20% increase in the weapon damage and other attributes. You can see changes in attack speed also.
So try fetching up more and more Materials (MK) to upgrade your weapon to max and takedown enemies instantly. Also keep your focus on upgrading the Core Module that is necessary to get more in Health, Stamina, and Battery.