fastest way to level upgrades?

[Puma Pants]

what's the fastest way to get upgrades level'd up?

[Tyyne]

Simple answer, play the game?

[InterWill]

Try to mine as many of the crafting resources as possible.
For Mastery Missions, if you get the bonus of the highest level you've cleared. But already at haz 3 you get 80% of the total reward (while being a lot easier than haz 5).

Biome Mastery has the biggest impact, since it gives bonus to all missions played on those biomes.
Sub Class Mastery is the second most important one
Weapon Mastery is the least important. They are still impactful, but you should prioritize weapons that you enjoy.

My tactic was:

First do GK2 Mastery for increased damage on Classic Scout Starting Weapon
Then Crystalline Caverns Mastery
Then Classic Scout Mastery
Then try to get some "easy wins" Weapons Mastery on Crystalline and/or with Scout (since I have those bonuses)

Prioritizing starting weapons like the LMG Gun Platform, Flamethrower etc

Then do Magma Core and Weapon Specialist Gunner, then clear some weapon's mastery that requires gunner and/or magma core

When you're in a position where you can reliably clear Haz 3 Masteries it starts going quickly. Just make sure to alternate between biome, class and weapon mastery so you get as much synergies as possible