Osten’s HW Relic Grind Guide

TL;DR:

Save your poetics for the umbrite grind (spend other currencies and/or do quests instead of spending poetics when you can)
Roll Need/Greed on loot while you're grinding dungeons in order to turn it into GC seals
Do as many ARR tribal quests that drop the tribal currency as you can, then spend the rest of your allowances on HW tribes
Start doing the Crystal Tower repeatable quest for aether oil early in case you get CT raids for the roulette, or in case you decide to grind CT for poetics
Do extreme primals for your Wondrous Tails to get the item drops for crystal sand
Don't overcap poetics or GC seals; if you're close to cap, read ahead and spend them on stuff you'll need later.

Gather one of each of six Luminous Crystals via Heavensward FATEs (each zone has a sub-100% chance of dropping one type of crystal)
Trade them in for an Astral Nodule (Fire/Wind/Lightning) and an Umbral Nodule (Ice/Earth/Water) with Syndony in Revenant’s Toll

Finish 10 dungeons with the i170 weapon from last step equipped. Can be done unsynced/undersized, and you can go through with a better weapon as long as you have the weapon equipped when you use the exit.
Snowcloak > Sastasha Hard > Qarn Hard > talk to NPC > Keeper of the Lake > Wanderer’s Palace Hard > Amdapor Keep Hard > talk to NPC > Dusk Vigil > Sohm Al > The Aery > The Vault

Need to acquire 10x of four types of item and 4x of another 4
Need 10: Unidentifiable Bone, Unidentifiable Shell, Unidentifiable Ore, Unidentifiable Seeds
Need 4: Adamantite Francesca, Titanium Alloy Mirror, Dispelling Arrow, Kingcake
See "Currency Calculations" below

Acquire 5 Aether Oil: 350 poetics or 1 run of all three Crystal Tower ARs each (15 runs total)

Need to get an amount of Umbrite and Crystal Sand. Maximum required is 80 of each but this is unlikely bc you can and usually will get lucky and need less due to randomness.
See "Currency Calculations" below

Acquire 50 Singing Clusters
You can spend poetics, OR
Do a weekly quest for 18 plus a daily for 1
Unfortunately you can only start doing these once you reach this step for the first time, but you can grind for additional relics' worth once you get here for the first one

1,750 poetics
but you’re better off doing the Crystal Tower raids because you need poetics for later steps and CT ARs give you poetics in addition to getting the necessary items without buying them

4,500-6k poetics, depending on luck (yes, really) (there’s no way around this one as far as I know) (sorry)
PLUS, some combination of:

3,750-5k yellow gatherer/crafter scrips (125 per 2 crystal sand)
?? number of HW battle levequests (possible reward from chest) (1 vilekin per 2 crystal sand)
90-120 battlecraft demi i + 60-80 battlecraft demi ii + 90-120 fieldcraft demi i + 60-80 fieldcraft demi ii (3+2+3+2 per 2 crystal sand)
150-200 moonstones/600-800k GC seals (yes, really) (5 moonstones/20k GC seals per 2 crystal sand)
30-40 inferno horn + 30-40 crag heart + 30-40 diamond tear - can be purchased from marketboard! (1 each per 2 crystal sand)
30-40 vortex feather + 30-40 barb of the whorl + 30-40 large levin orb - also purchaseable! yay! (1 each per 2 crystal sand)
a whole pile of poetics, i’m not even gonna calculate this, dont use poetics for this step i s2g you still need more later
30-40 of each unidentifiable from earlier, but also dont use these if you’re doing more than one weapon unless you have hella extras, they’re enough of a pain to get for the first step

Anywhere from 0 to 2k poetics depending on how patient you are
Ideally, spend no more than 1,240 poetics here—you can get 19 of the quest item as soon as you unlock this step which accounts for the other 760 poetics
If you’re doing multiple weapons, make a point to do the daily/weekly quest that gives you the requisite item every day/week to save time and poetics later

Step 7: some combination of up to 1,500 poetics and up to 60k GC seals; pneumite costs 100 poetics or 4k seals each and you need 15
Step 8: 500 poetics