Upgrade weapons at the main hub's left-side workbench. You reach this hub after beating the first boss, and you can buy some upgrade materials there with Catalyst. To upgrade weapons in Valor Mortis past those early purchases, you'll need the blue whip power first; it opens the routes to the quarry and church roof materials.

How do you upgrade weapons at the main hub?
Interact with the workbench, select your weapon, and spend the material listed under Upgrade Requirements on the upgrade you want. Check the effect before committing: damage increases and special effects serve different purposes.
For example, the Rapier's Remise makes consecutive parries increase your next Heavy Attack's damage. Attacking enemies prolongs the effect. It's worth it if you like turning parries into big heavy hits, which means your parry timing needs to be reliable.

If you're short on Crude Blade Material, buy it at the hub and return to the upgrade menu. The shop's Buy tab lists it as a material used to upgrade blades. Stock is limited, so choosing where to spend it matters.

You cannot recover spent materials or cancel weapon upgrade levels. Try a weapon before you sink materials into it, especially if you're torn between the Saber and Rapier; both are compared in the weapon list. You can switch between alternative upgrade effects for free at the abandoned church, so experimenting with an unlocked effect is a much smaller commitment.
Which pistol upgrade should you choose?
Choose the round-return effect for ammunition recovery, or the melee-damage effect to reward shooting an enemy before hitting it. Both connect gunfire with melee attacks. No pistol yet? Here's where to get it.

Resourcefulness: recommended for recovering ammunition
Using a Melee Attack on an enemy you've shot extracts a single Pistol round. This activates only once per enemy. Pick it if you want an immediate ammunition return when mixing shooting and melee; repeatedly hitting the same enemy will not keep giving rounds back.
Melee damage after shooting: choose it for a stronger follow-up
The alternative increases your next melee attack's damage based on how many rounds you've fired into that enemy. Choose it when you want gunfire to set up a harder melee hit. You can return to the abandoned church and switch effects without spending more materials if the rhythm doesn't suit you.
Where is Pure Blade Material in the Valley of Ash quarry?
Pure Blade Material is at the top of the quarry's blue grapple route, reached by taking the elevator from the Quarry lantern and climbing the movable platforms. Get the blue whip power in the village first, then backtrack.
Travel to the Quarry lantern
Select Quarry in the lantern travel menu. The region heading reads Valley of Ashes.

Turn around and take the elevator up
From the lantern, turn around and use the elevator. You should be heading toward the wooden structures above the quarry's stone walls.

Grapple beside the pulley
Position yourself beside the pulley and use the blue grapple points. Grapple again, then pull the movable platform down. The hanging structures may look like background machinery at first; the blue points mark the parts you can use to climb.

Ride the pulled-down platform
Jump onto the platform so it carries you upward. You can jump past the first row of enemies rather than stopping to fight everyone on this climb.

Pull the upper platforms and reach the musketeer
Pull down the next movable structures and jump up toward the musketeer. Watch your back: another musketeer can shoot you from behind, so lingering on the climb is risky. When you target the upper blue point, the Pull (Hold) prompt appears.

Collect Pure Blade Material at the top
Finish the climb and collect Pure Blade Material on the upper route. You're now on the wooden ledge against the quarry's pale stone wall.

How do you reach Distilled Blade Material on the church roof?
Reach Distilled Blade Material by using the blue whip to climb the church exterior, jumping through the broken section, and continuing to the backside. The church roof route requires the blue power, so return after unlocking it.
Leave the hub toward the first boss and circle the church
Go through the main hub door that leads back toward the first boss. Run around the church and look for breaks in its exterior, then use your blue power to pull yourself up.

Continue around and jump through the broken section
Follow the route around the church, then jump through the broken wall section. Look for the large opening with a wooden plank projecting from it.

Reach the backside and collect Distilled Blade Material
Continue to the backside to collect Distilled Blade Material. The elevated church walkway overlooks the interior below.

The church roof material is Distilled Blade Material. For Pure Blade Material, take the quarry grapple route instead.
How do you get Superior Firearm Parts through the small opening and hidden wall?
Destroy the hidden wall by targeting it through the small opening, then go around the right side to collect Superior Firearm Parts.
Look through the small opening and destroy the wall
At the small opening, look through to the wall beyond and destroy it. Stay focused on the wall behind the gap rather than trying to squeeze through the opening yourself.

Go around the right side
Once the wall is destroyed, circle around the right side to the opened route. The outdoor path runs beside the large growth-covered structure.

Pick up Superior Firearm Parts
Collect the parts beyond the opened wall. The pickup notification reads Superior Firearm Parts.

Take the materials back to the hub workbench and put them into the weapon you plan to keep, then carry on with the walkthrough.