I would like to acknowledge 南云归风's equipment guide as a cross-check source for many of the equipment entries:
https://test-steamproxy.haloskins.io/sharedfiles/filedetails/?id=3160614270
Much of this guide's structure was inspired by KH/TWRoxas' great equipment/KARON guide for Revision; check it out if you're looking for base game information without having to parse the JP wiki through Google Translate.
https://test-steamproxy.haloskins.io/sharedfiles/filedetails/?id=2889115638
The original GROW diagram is sourced from an Anon on dcinside's Astlibra Minor Gallery[gall.dcinside.com]; I've only translated the Hangul notes back to English and did rudimentary typesetting in turn even if sharing this diagram was also the biggest driving force behind creating this guide.
Special thanks to Inarrou for helping me fill in gaps with the Small Metal rewards.

As usual, there will be major spoilers throughout the guide.

The equipment sections will briefly indicate all of the gear available in Gaiden, where to find them, and their corresponding mastery reward (or special effect, if any for accessories).
Equipment functions almost identically to Revision: weapons, shields, and armor gain proficiency as you kill monsters inside the cave, and will either grant a POLIN skill or have their stats boosted alongside producing an item (typically a Magic Crystal). Any mastered equipment can be re-equipped to use its POLIN skill at no Magic Crystal cost while that gear is equipped.
Unlike Revision, equipment enhancement can be performed very early on in Gaiden instead of being relegated to the Postscript. It should also be noted that Postscript Boards are relegated to your Styles and are overall more simplified compared to Revision's approach.

All Master entries with an "EP:" denote Enhanced Performance equipment.

1. Divine Sword Ec Stasia requires 20 Primordial Fragments and an Alexandrite to craft.
   Primordial Fragments only drop from the cell enemies that populate Stratum 10. They drop fairly frequently, so give it a run or two if you're short fragments.
   To obtain the Alexandrite, you need the Great Sage's Rope and Iron Bars Key from mastering the Great Sage's Robe and Kutune Shirka. Go to the well past Rispadar's clothing store and use the Great Sage's Rope so you can enter the well. Go all the way to the left (ignore the bottom path for now) to reach the Rispadar prison from Revision chapter 6. Use the Iron Bars Key to open up the Hero's old prison cell. Interact with the bed there to fight Anubis' 3rd phase (in its chapter 8 form).
   Defeating Anubis will drop Soul Balance Collibra...as a key item. Go back to your mother in the bakery and she will barter it in a cutscene for an Alexandrite (all but said to be with the Hero). Return to Mokyun in any Postscript Stratum to craft Ec Stasia.
   
2. The cutscene will trigger after traversing at least one stage in the Cave at the beginning of the game, whether you return safely or die.
   
3. Legion still resides at the bottom of Rispadar's well as in Revision. You only need the Great Sage's Rope from the Great Sage's Robe to access Legion; simply fight through the well's bottom path gauntlet as before and defeat Legion to obtain Ignition.
   
4. Mastering Elna Saga will produce a Sun Stone. The Sun Stone can be traded to Komugi for a Herb, which can be synthesized with Gunpowder to produce chapter 7's Sleeping Aid (note that you need Alchemy Recipes from 44 Small Metals to do this).
   You can then make the guard blocking the way to Rispadar's palace fall asleep with the Sleeping Aid to get through. Hit the target then Air Surf all the way to the other side to obtain Faxana-du in the chest.
   
5. You will not get another chance to fight Kai if you lose, so be sure to reload to before the fight if you want the Kaiser Ring!
   If you're struggling against Kai, try using a magic build to whittle down his health with Dark. You can try to pick off his orbs with Cutter as well. Otherwise, he has the exact same attacks as with his appearances in Revision; don't be too greedy with generating ST on hits if you're on the Magician style.
   
6. This triggers after clearing Stratum 3 and being told to go to the inn.
   
7. This is blocked off by a door opened with the Goblin's Stone Plate. The upper Stone Plate Fragment is rewarded with 32 Small Metals, while the lower fragment is rewarded by mastering Murasame. Note that Alchemy Recipes is also required to synthesize the completed stone plate.

Unlike Revision's GROW tree consisting of chapter-locked zones with varying shapes, Gaiden's GROW tree is simplified down to a massive grid with 8 rows/branches corresponding to each stat.
From top to bottom, these branches add to HP, ST, ATC, DEF, MAG, SPD, LUC, and ADA. As before, HP, ATC, DEF, MAG, and SPD will gradually ramp up their stat increases further down the tree, while ST, LUC, and ADA's per point values are static throughout the DLC.
All text written directly on the tree correspond to equipment treasure chests.
The only other commonality with Revision's GROW tree are the center locks, which will disappear after entering the 6th Stratum for the first time in Gaiden's Postscript half.
As noted on the image, the far right locks will unlock for each God defeated in the Arena.
The last point of interest is the upper-right corner at the end of the HP/ST rows: these are the infinite GROW stat nodes, which also function in exactly the same way they did at the bottom of Revision's Postscript section of the GROW tree.

The POLIN Skills system is functionally identical to Revision's KARON skills and many of its skills (both base and Board swaps) make their return alongside a handful of new or remixed skills. A few Revision skills, such as Swordsmith or Sub Weapon, have their functionalities replicated by Gaiden's Styles.
Unlike Revision, boards in Gaiden cannot roll pieces that replace skills. What you see here is what you get.

• Ranged Master: Cuts the animation lock/reload time of sub weapons by about 1/3. This also affects their attack interval and does stack with LIBRA's Use Speed.

• The Art of Thievery: Drop rate increases by about 4-5%; doesn't affect chest drop chance.
  
• Polin's Nose: This skill's map refers to the minimap while unpaused; not having this skill won't affect the Handmade Map Kit's paused maps.
  
• Sturdy: Equivalent to 20 ADA, which grants about +7 to status resistances.
  
• Digging Master's Focus: Prevents low hit rolls (accompanied with a dull "kang!") when mining.

• Herculean Power: This still makes Kirihide travel farther by one MC's width worth??
  
• Gun Hammer, Sword Energy: Both will stack with each other and appear to scale based on your magical power, as do other weapons' innate sub-attacks (e.g. Thunder Sword, all staves).
  The Thunder Sword will not proc Sword Energy, though Ec Stasia will as of v1.0.4.
  
• Shuffling: Roughly half your walk speed. Also applies to sub weapons' animation locks.
  
• Berserk: A bleed tick will count as a "hit" to kill. Dying Curse and HP burn from Darkness don't count.

• Recovery Guard: Heals about 20% of your max HP. Scales with LIBRA's HP Recovery.
  
• Step: Usable in the air; Air Surf will take priority if using the Dash button, but double-tapping left/right will work just fine if you need Step specifically. And while Air Surf can't be used when walking off an edge, Step will still work.
  Has about a second cooldown before re-use. Using Air Surf seems to freeze that cooldown timer during Air Surf.
  Equip Eternal Galla Shin if you want both Dash and Step as they conflict otherwise.
  
• Second Wind: The iframes extend to about half a second. It's very noticeable if you stand on lava/spikes or get hit by a laser.
  
• God's Left Hand: The ("intentional") loophole from Revision still works:
  
  1. Equip Wicked King's Shield to get GLH for free with a 1 or 2-range weapon equipped
     
  2. Equip whichever desired 3-range weapon
     
  3. Swap to your desired shield
     
  4. Repeat steps 1-3 if swapping to another 3-range weapon

• High-Speed Incantation: Appears to roughly halve the default pre-cast delay of half a second.
  
• Great Magic: Stacking this with Dragon Cup will increase the ST drain grace period to about double its normal time.
  
• Absolute Territory: Appears to be a negligible increase from the default 1s iframe, if at all from testing by casting different spells on top of lava/spikes then frame counting recordings with/without the skill.
  If the NGE reference wasn't clear enough, this is outright called 絶対領域 in JP.
  
• Flawless: Increases ST costs by 10%. Not nearly as useful given the lack of transform-type magic compared to Revision and the above 1s iframe.
  
• Darkness: HP cost is 3x the spell's cast cost without considering Full Burst nor Artisan in exchange for a 60% spell power buff. If you don't have enough HP to pay for a spell, you won't be able to cast it.
  
• Eliminate: Increases ST costs by 10%.
  The ST cost increases of both Flawless and Eliminate are additive with each other.
  
• Demon Rage: The 2x ST cost will not affect Flawless nor Eliminate's cost increases.
  
• Full Burst: The spell power buff is 0.8 x (amount of ST used / original cast cost). It's completely possible to do less damage with FB if you don't use more than 1.25x a spell's original ST cost.
  Darkness, Demon Rage, and Full Burst's spell power multipliers are all additive with each other.
  
• Artisan: The ST consumed will always be 5% less of the final cast cost after considering Flawless, Eliminate, and Demon Rage if they are active.

• Origin: Appears to be a roughly 15% multiplier?
  
• Maxwell: Stacks additively with Ifrit-Luna. Appears to be about a 20% multiplier.

• Gumption: Damage Cut 10% => 15% at 20 combo, Range 1.2 => 1.3 at 50 combo, move speed 1.3 => 1.4 at 100 combo, Steal Gold at 150 combo and Force Fever at 300 combo are buffed by roughly 20% (denoted with another +)
  
• Magical Power Conversion: Unaffected by LIBRA's HP Recovery.
  
• Gemstone Power: Damage multiplier appears to be (Corundum owned / 2)%. This affects all damage inflicted, physical or magical.
  As implied by the tooltip's phrasing, Gemstone Power is multiplicative with all other damage multiplier sources, including Berserk for physical attacks and Darkness, Demon Rage, and Full Burst for magic.
  
• Divine Reversal: The final boss has two attacks affected by this: Breath of Death, and Breath of Life. Normally, Death hits extremely hard and will likely one-shot you, while Life just heals you. This skill reverses the breaths both ways so Death isn't fatal but Life will kill you. Unequip this as needed during the fight.

The Arena will unlock after clearing the 1st Stratum and the following sequence with the Scales. Unlike Revision's Arena, the Gaiden Arena battles are replays of Revision's main bosses; other than all bosses having scaled up stats (including Anulis now that having Polin skills is taken into account), their behaviors remain the same as before.
There are also no hard progression locks between Arena battles beyond the obvious stat spikes every few fights. Thus, Arena progression can be roughly tiered from weakest to strongest as follows:

1. Early first half: Golden Dragon, Terra-Behemoth
   
2. Mid first half: Agni, Meldine
   
3. Late first half: Deus Ex Machina, Marionette
   
4. Beginning of Postscript: Anulis, Mira-Bahamut
   
5. Mid Postscript: Gleipnir, Pure Silver
   
6. Endgame: God fights, The Barkeep

Clearing each fight will unlock the next in the sequence, excepting Pure Silver unlocking the six God fights. These Gods can be fought in any order, but all of them must be defeated to challenge the Barkeep.
As with Revision, Arena battles will produce a reward on the first clear. These rewards are as follows:

You will be able to start showing Small Metals to Meloo once you're prompted to visit the inn at the end of the 3rd Stratum. She will start giving rewards every 2 metals from 10, all the up to 58.

Your Little Sister's Hobby
Gaiden: Get the Handgun from your little sister.
The Handgun requires a Slime Jelly and a Penguin's Egg to purchase from Komugi, both which only drop in the 6th Stratum. Consequentially, you won't be able to get the Handgun until early in Gaiden's Postscript.


Dummy Breaker
Gaiden: Destroy the dummy in the backyard.
The dummy has defensive stats on par with mobs in the later parts of Gaiden's first half. I recommend attempting to destroy it when you can reasonably unlock Zeus at the 4th-5th strata.
Destroying the starter dummy will drop the Level Emblem and spawn in Revision's rock dummies in its place.

Lottery Master
Gaiden: Collect all the grand prizes from the Luxury Lottery.
Mokyun's luxury lottery contains +1 Trinket Slots as its gold ball prize. Collecting all 5 of them will unlock this achievement and replace them with 3 Magic Stones.


Scales' Latent Power (Gaiden)
Gaiden: Strengthen the scales once.
Once you've cleared the 11th stratum, go back to the Yggdrasil interstice in Rispadar and talk to the old lady there. She will unlock scale strengthening at this point.
Unlike the main game, each scale strengthen only costs 20k gold.

A Talented Older Sister
Gaiden: Acquire all Styles
Gaiden has 15 Styles to obtain: Explorer, Swordswoman, Magician, Guardian, Thief, Barbarian, Knight, Lancer, Dancer, Ninja, Samurai, Heavy Warrior, Assassin, Witch, and Hero's Reflection.

• Delivering bread to the clothing store will unlock Explorer.
  
• Swordswoman is inside the weapon store's chest. Use Air Ride to reach it.
  
• Magician is found in Stratum 1 midpoint's chest.
  
• All other styles can be found at a very low chance from chests inside the cave. As far as I know, the styles from Guardian to Ninja can be found in the first half of the game, while Samurai to Hero's Reflection are exclusive to the Postscript half.

A Magical Older Sister
Gaiden: Learn all magic spells.
Below are all of the spells unlockable in Gaiden. Note that spells beyond the fourth row (or fifth in Fire's case) are only accessible in Gaiden's Postscript half. You will also need to defeat Gods in the Arena to complete this achievement; see the GROW tree for more details.



A Stylish Older Sister
Gaiden: Acquire all outfits.
Simply purchase all outfit items from the clothing store. Without accounting for Golden Ticket discounts, this will cost you up to 300k gold.



Indirect Kiss
Gaiden: Acquire the Lollipop.
The Princess sells the Lollipop for 99 tissues.
If you don't want to gamble all of your normal lottery tickets for tissues, get Alchemy Recipes from Meloo with 44 Small Metals. This will unlock recipes to convert a (luxury) lottery ticket and a Refining Kit into a tissue.

Helpful Dog
Gaiden: Learn all POLIN skills.
Refer to the Polin Skills section for more information. Note that unlike Revision, all of Gaiden's unlockable skills come from mastering equipment.



Satisfied With Just Looking
Gaiden: Show all the small metals to Meloo.
Talk to Meloo again after obtaining the Mysterious Blueprint for 58 Small Metals to make this achievement pop up.



Champion of the Arena
Gaiden: Dominate all the arena battles.
See the Arena section for more details on how Gaiden handles the Arena.
Remember to talk to the barkeep after beating him to make the achievement pop up!

My Beloved Older Sister
Save your little sister, while saving the town in the process.
Defeat Geist following the 5th Stratum's boss. This achievement will pop up at the end of the epilogue sequence.



Two Heroes
Gaiden: With two of them around, everything will be fine.
Defeat the final boss at the Lowest Stratum following the 11th Stratum. This achievement will pop up after the post-credits cutscene.



Kaiser
Gaiden: Go against destiny and defeat Kai.
Missable achievement!
You will fight against Kai right after the 2nd Stratum's boss. You will not get another chance to fight him if you lose, so be sure to reload to before the fight if you want the achievement! He will also drop the Kaiser Ring if you win.
Refer to the 5th Equipment footnote if you're struggling with winning against him.

Side Story Master
Gaiden: Do absolutely everything in Astlibra Gaiden.
Despite the achievement's description, this is not actually a 100% completion achievement!

1. A final boss time trial orb will appear at the Lowest Stratum after defeating the final boss. Clear it once to unlock the final boss Leveling Challenge orb.
   
2. Clear this Leveling Challenge twice to unlock this achievement and the Golden Cogwheel once you return.

May 03, 2024: Clarified accessing the well by indicating the need to use the Great Sage's Rope.
Mar. 02, 2024: Modified Sword Energy footnote in light of v1.0.4's hotfix.
Feb. 24, 2024: Fixed poor/redundant wording for getting Ignition and introducing Gaiden's Arena.
Feb. 22-23, 2024: Added notes for select POLIN skills, using the JP wiki's KARON page as a main reference.[w.atwiki.jp] Looks like many of Revision's quirks/jank tied to returning skills is still intact in Gaiden.
Feb. 21, 2024: Guide published.