You'll be getting the upgrade materials as drops from defeated enemies and from resource nodes. You'll notice drops as the orbs that come out of enemies when they die and nobes as a shiny light on the ground, green orbs are common or uncommon items, while yellow/golden orbs contain rare items. These drops remain on-screen for a while but do eventually disappear, so you might want to prioritize moving towards them to automatically pick them up even if you're still in combat. You can also use Words to increase your drop rate. Words featured in the 'mah' group all act to increase drop rate chance, which levels up the chances of a drop and the chance of that drop being rarer. Consider heading into the word edit menu to change up your words in this direction before hitting the grind.

*Some say that the item drop rate words actually decrease the chance for the rare materials to drop but nothing is confirmed 100% so if you struggle to find a rare material I recommend switching it up and seeing what works for yourself. I have tried both methods and haven’t notice any big difference but it can vary for every one because it is based mainly on luck.

**It is not 100% confirmed that setting the game to hard will boost the material drop rates, and if it does sometimes it’s not worth it. It takes a lot of extra time to actually defeat some enemies and farm the materials compared to the normal mode. Think about it, even if you get like a 5-10% extra drop rate (on hard) you will farm with a slower pace probably double or triple the time you do compared to normal, so even if you have a little better drop rates you dont farm fast enough. We don’t know the actual stat numbers so I can’t actually tell which is better, I can only tell you that I have not seen much of a difference between hard and normal, but you can try for yourself (same with words) and see what works best for you. The Hard difficulty is great to use when farming Memory Alloy (check the Memory Alloy section).

Junk Heap Level B2. Blow up a barrier to take the northern path, you will find a room filled with crates which they have a chance to drop this rare material, among some others.

Seafront Beach - available as a natural gathering drop in the area. Its a rare material so just pick up every drop you see and you will get them after some attempts.

Drops from Robot enemies in the Junk Heap.

Drops from Robot enemies in the Junk Heap.

Dropped by flying robots in the Junk Heap.

Can drop from the adult magic-casting Shades that are found in the Northern Plains and atop the Lost Shrine in the later part of the game. Enter and exit the boss room to force respawns.

Dropped by the basic boxy robots in the Junk Heap or purchased from the Two Brothers shop outside of the Junk Heap.

Drops often from the flying robots found throughout the Junk Heap dungeon.

Available from a gathering point, but only rarely - it shows up at a spawn directly beneath the house of the Aerie Chief, and on the path leading out of the Aerie back to the Plains.

1st method: One thing you can do here is use the Aerie’s nature to cheese it - if you don’t get your desirable drop, you can keep jumping into the abyss until you die. When you respawn, the resource node will be back… giving another chance to get the egg.

2nd method (By Azure Fang written in the comments of this guide):

You can refresh the looting spots by saving and loading from the mailbox in the Aerie.

1) Enter Aerie and grab/check both spots. (One under the Mayor's house & the other at Kaines house)
2) Save and reload.
3) Seconds after load, the spots will populate (or not, depending on RNG). Check both spots.
4) If neither spot populated, just reload. If one or both spots had materials, save then reload.
5) Repeat steps 3 and 4 until your fingers bleed.

Given that you can see the Egg spot glowing from the first platform, this is significantly faster when farming both materials (Egg/Broken Pottery) simultaneously.

Rare drop from the ''Hat Shades'' in the Southern Plains.

A rare drop from the armored shades that carry the large tower shields. These can be found in both the Northern and Southern Plains, but their spawning is also partially governed by the in-game weather; in sunny but mildly overcast weather, more spawn than when it is completely overcast. Additionally, you can find some of these Shades in the Shadowlord’s Castle during your story-based trip there.

Obtained from harvesting in most areas or purchased from the Materials Shop in Facade.

A moderately uncommon drop from the larger types of Shade. Part of a series alongside the Simple Machine (common) and Elaborate Machine (Rare). These appear all over, including in the Barren Temple, the Lost Shrine if you return there right before the final battles, and on the Southern Plains approaching Seafront town.

Alternatively, go to the Lost Shrine, where two of these enemies ambush you as you climb the tower. Exiting and re-entering forces a respawn

Finally, if you complete the quest A Magical Stone, which is only accessible at the very end of the game after collecting all five fragments but before heading for the final battle, you can talk to the Fortune Teller lady to replay Nier’s prologue. The enemies here can be farmed for all three machines.

You’ll need a lot of these for weapon upgrading. It’s found commonly at various resource deposits throughout the world, including on the Eastern Road - though to be honest, the better bet is to buy them from Two Brothers’ Weaponry outside the Junk Heap.

Crystals can be gathered from resource deposits in the Desert and Barren Temple, but they are significantly rarer drops. However, you’ll only need a handful. You can also buy them from the Facade Materials store for 2400 Gold.

Obtained from crates on level B2 in the Junk Heap.

A rare drop from the smaller shades out in the world. A good farming spot is in the 2nd Basement Floor (2B) of the Mansion; in one of the first rooms here 40+ of these shades spawn in waves. Because it’s near the stairs, you can kill them, go up a floor, then come back down, forcing a respawn. Dark Whirlwind makes this easy work, too.

1st method: An rare drop from a resource node that’s found underneath the Mayor’s house in The Aerie. One thing you can do here is use the Aerie’s nature to cheese it - if you don’t get the Eagle Egg drop, you can keep jumping into the abyss until you die. When you respawn, the resource node will be back… giving another chance to get the egg.

2nd method (By Azure Fang written in the comments of this guide):

You can refresh the looting spots by saving and loading from the mailbox in the Aerie.

1) Enter Aerie and grab/check both spots. (One under the Mayor's house & the other at Kaines house)
2) Save and reload.
3) Seconds after load, the spots will populate (or not, depending on RNG). Check both spots.
4) If neither spot populated, just reload. If one or both spots had materials, save then reload.
5) Repeat steps 3 and 4 until your fingers bleed.

Given that you can see the Egg spot glowing from the first platform, this is significantly faster when farming both materials (Egg/Broken Pottery) simultaneously.

A moderately uncommon drop from the larger types of Shade. Part of a series alongside the Simple Machine (common) and and Comlex Machine (uncommon). These appear all over, including in the Barren Temple, the Lost Shrine if you return there right before the final battles, and on the Southern Plains approaching Seafront town.

Alternatively, go to the Lost Shrine, where two of these enemies ambush you as you climb the tower. Exiting and re-entering forces a respawn

Finally, if you complete the quest A Magical Stone, which is only accessible at the very end of the game after collecting all five fragments but before heading for the final battle, you can talk to the Fortune Teller lady to replay Nier’s prologue. The enemies here can be farmed for all three machines.

This rare drop can be found in a few places, but the easiest location to gather Fluorite is in The Junk Heap dungeon. Head down to the Basement Level 2, B2F - you’ll want to blast away a barrier with a bomb to access a northern pathway. In a large room here there’s a bunch of crates - and when you smash them, they can drop Amber, Fluorite, or Damascus Steel. You can reset by exiting and re-entering the area.
Alternatively, you can also complete the ‘Disturbing the Sleep of Kings’ side quest, which is required to get a weapon, and then return to the Barren Temple. You can now head towards the boss area - on the bridge here, there’s resource nodes that have a decent chance of dropping Fluorite, as do the other nodes in the temple. Beware, however - not every node in the temple spawns every time you load the area - only some of them.

This material is a rare drop from the adult mage-like Shades that are heavy on the magic. You can find these enemies in the Northern Plains (weather dependent - it needs to be overcast), and at the top end of the Lost Shrine, right before the boss room. Enter and exit the boss room to force respawns.

In the Northern Plains, they’re scattered about - by both bridges, near the boar, and some generally roaming the plains. You can force respawns by activating and failing the sheep mission ( jump off to fail) located in the eastern part of the map (I think this mission is avalable only after you collect all the pieces of the key but Im not 100% sure). If the mission is not available you can leave and enter Northern plains to force respawns, keep in mind that the weather matters too so some times of the day these shades do not spawn.

You can get Giant Eggs from resource nodes that are scattered around your home village. You can reload the area to force respawns if you need.

Uncommon item dropped by goats. You can farm 4 goats in the Eastern Road and easily respawn them by leaving and entering the area.

You can get this rarest of the ore-based resources from the Desert, Aerie, and Barren Temple all from resource gathering nodes. I liked to farm it in the dessert because you can reset the nobes by entering and leaving the Barren Temple. Unfortunately, this ore isn’t available for sale from the weapon upgrade shop either, thanks to its rareness.

Iron is one of the more common ores, but you’ll need a number of them for upgrades. They’re found at a lot of resource nodes right across the world, from the Eastern Road to the Desert, Aerie, and more. It can also be purchased at the weapon upgrade smithy for 1000 Gold.

This is one of the rarer drops from the enemies of the Junk Heap. It more commonly drops from the stronger robots.

it’s a rare drop from the flying robots in the Junk Heap. It is one of the most significant items you’ll want, as this material is required to complete a side quest that locks away one of the weapons. But you’ll also need some for upgrades.

Memory Alloy is the rarest Junk Heap dungeon material and it drops only from P-33 robots (Beepy) that you encounter your way through level B2 . There’s only 4 of those in the whole dungeon so you’ll just need to grind your way through level B2 Repeatedly. Its a little pain to farm them but you dont need a lot of them, propabliy about 10-12 to upgrate all your weapons.
Lucky for you there is a way to make this a lot easier: Set the difficulty to hard. Skip the first robot on the B2 floor. Go on the rail-cart, and in the very next room a robot will spawn. Kill it, and see what drop you get. If you dont get a Memory alloy, let the remaining small drones kill you. You will respawn just outside the room. Repeat until you get an Memory alloy. When you get that one, go ahead and go to the next room to the south where 2 robots spawn. Kill them and see if you get an Memory alloy, if not, die. Repeat.

You’ll need lots of these for upgrades. They’re a rare drop from the flying shades that spawn in the desert after you complete your second visit to Facade (with the wolves) in the second half of the game. There are three spawn locations one near Facade’s entrance, another one inside the Sandstorm, and one near the stairs leading to the Fast Travel boat.

*Bonus: After you complete the side quest ''The missing girl'' in Facade you can access another side quest located in the dessert called ''Shadows of the Desert'' (It appeared to me only after I obtained the Facade key fragment). You will have to find the girl and talk to her three times (every time you find her you must leave the area to change her spawn location), after that Shades will start attacking and you will have a chance to kill 5-6 of the flying ones. I personally found a metal piercing by completing the mission, but even if you don’t find any you still get rewarded with 30000 gold.

The meteorite weapon upgrade material is another one found on a resource node out in the world - it’s one of the rarer drops specifically from resource nodes in the Desert.
Meteorites can also drop from the nodes inside the Barren Temple. For your best chances, complete the ‘Disturbing the Sleep of Kings’ side quest to unlock the whole dungeon again. On the bridge to the boss room, there’s two resource nodes out there, plus some scattered across the other dungeon rooms.

To get Moldavite, you’ll need to scour and constantly reload the Desert and Barren Temple gathering points - or you can buy it at the Seafront Materials store, but it’s eye-wateringly expensive.
Don’t forget, in the Barren Temple if you complete the ‘Disturbing the Sleep of Kings’ side quest, you can access more of the dungeon again including the bridge to the boss room, which has two resource nodes.

A more simple one - the Mysterious Switch loot drops from the flying robots found in the Junk Heap. It’s uncommon - so not the most common, but not as rare as some others.

Obtained from harvesting in either Seafront or the Southern Plains, but is also available for purchase fairly cheaply (700 Gold) at the Seafront town materials shop.

Obtained from harvesting in Seafront or can be purchased from the Material Shop in Seafront.

You’ll need a number of Pyrite materials - and it’s another one that unfortunately is found at resource gathering nodes in the Desert overworld area and also the Barren Temple dungeon.

The Rusted Clump is a relatively unique material for upgrades, it can only be found as a random-chance drop from fishing. Ι’ve seen them all over Seafront, the Northern Plains, the Eastern Road. Use a lure, and every so often you’ll fish up some handy junk.

*You can also farm them in The World of Recycled Vessel DLC were you play as Father Nier as it is a common drop (see more at Recycled Vessel DLC section).

Can sometimes be dragged up by fishing as a junk catch instead of an actual fish. This can happen in any place where you fish, and you should use a regular lure, the most basic bait, to minimize how many regular fish you catch.

A rare drop from small shades roaming around the Northern Plains. Also dropped from the small hat wearing shades in Southern Plains & Shadowlord's Castle.

Scorpions Claws are the most common drop from Scorpions, which can only be found in the desert. They spend most of the time under the sand and will only jump up and attempt to poison you once you are really close.

Obtained from harvesting in most areas or purchased from the Two Brothers shop outside of the Junk Heap but it doesn’t come cheap, it’s sold for 1500 Gold each.

This is one of a set alongside the Complex Machine (Uncommon) and Elaborate Machine (Rare) - and those rarities makes this the common, easiest of the trio to get. They’re dropped by the larger types of shade you fight commonly - the taller, lanky, adult-sized ones. There’s a few locations you can find these including the Barren Temple, the Lost Shrine right before the end of the game, and on the Southern Plains en route to Seafront town.

Alternatively, go to the Lost Shrine, where two of these enemies spawn as you climb the tower. Exiting and re-entering forces a respawn.

Finally, if you complete the quest A Magical Stone, which is only accessible at the very end of the game after collecting all five fragments but before heading for the final battle, you can talk to the Fortune Teller lady to replay Nier’s prologue. The enemies here can be farmed for all three machines.

This is a rare drop from the shades that use swords that are found both in the Underground Shrine and also in the Lost Shrine. Keep in mind they spawn less often in the Lost Shrine later on in the game when the enemy presence there elevates. For farming purposes, the multi-floor underground lab is arguably best for these, as you can farm the enemies, then switch floors to force respawns.

Dropped by ground-based robots in the Junk Heap or purchased from the Two Brothers shop outside of the Junk Heap.

The Subdued Bracelet material is a pair with the Metal Piercing item, in that it drops from the airborne shades that appear in the Desert in the later parts of the game, after you’ve finished the story associated with your second visit to Facade (that’s the storyline with the wolves). The flying shades only spawn when you get near their spawn points, which are near Facade’s entrance, inside the Sandstorm, and near the stairs leading to the Fast Travel boat.

Dropped by ground-based robots (rare) or from large robots (P-33 robots) on level B2 (common) in the Junk Heap.

Drops from the armor-clad shades found in the Lost Shrine and Aerie, and can be handily farmed during the later stages of the game when the Lost Shrine is teeming with enemies.

After finishing the game for the first time (ending A) you unlock a ''mini game'' were you play as Father Nier. You can access it by interacting with the library shelf at the first floor of your main house, there are 3 challenges but in order to access the 3rd one you must first complete the other two. After you unlock the 3rd challenge you can replay it in order to farm some random materials, experience and gold. Keep in mind that the materials drop only at the 3rd level of the 3rd challenge (the level in Emil’s lab) brake all the boxes to farm the materials.
Here is what materials i got after some attempts:

Amber, Copper Ore, Eagle Egg, Gold Ore, Iron Ore, Meteorite, Moldavite, Pyrite, Rusted Clump, Silver Ore, Titanium Alloy

*The most common ones for me were Copper Ore, Iron Ore, Rusted Clump, Silver Ore and Titanium Alloy.

Keep in mind that the drops are random, you wont get the same materials every time. You will also get 50000 gold and 10000 exp every time you complete the challenge so this is a great way to farm gold too (it takes only 10-15 minutes to complete).

Here is the link of the original guide were i got all the information that i didnt know myself by just playing the game.

https://www.rpgsite.net/feature/11032-nier-replicant-upgrade-materials-where-to-get-all-items-for-weapon-upgrades