First of all, with base DMG, I mean the damage that a valkyrie deals with her skills which have their own multipliers. You can find the base damage value of a valkyrie using the following simple formula:
ATK × skill that deals %ATK of Physical/Elemental DMG = base DMG

Void Drifter will be assisting me here.
So, Void Drifter's ATK value is 1094. Her combo attack, Void Flechette, with some help of a passive skill, deals 320% ATK of Physical DMG.
Thus: 1094 × 3.2 = 3500 which means Void Drifter without stigmatas and weapon bonuses will deal 3500 points of Physical DMG to a target. This is her base DMG.

Let's get our Void Drifter ready for an actual fight:
● Schrodinger (T): +205% Total DMG;
● Dark Jixuanyuan (M) and (B): +40% Physical DMG;
● Tranquil Arias: +50% Total DMG, +15% Physical DMG;
● she has passive bonuses: +50% Total DMG and +15% Crit DMG;
● she is MECH-type and will be attacking a BIO-type enemy: +30% DMG.

Physical DMG multiplication formula (built-in bonuses, stigmatas, weapon, type advantage/disadvantage):
(1 + Total DMG) × (1 + Physical DMG) × (base 200% Crit DMG + extra Crit DMG) × (1 + type advantage/disadvantage or ±0.3)
↓
(1 + 2.05 + 0.5 + 0.5) × (1 + 0.4 + 0.15) × (2 + 0.15) × (1 + 0.3) = 17.54

Finding out how much damage Void Drifter will deal when fully geared:
Base DMG × final multiplier
↓
3500 × 17.54 = 61390

Important to know: Elemental DMG represents every element (Ice, Lightning and Fire), so you can use Elemental… uh… something with a valkyrie of any element (Physical is not an element, obviously).

Hawk of the Fog will lend me a hand here.
My Hawk of the Fog has 1372 ATK and her combo attack, Kei: Taranic Talons, with some help of a passive skill, deals 430% ATK of Lightning DMG.
Her base DMG will be…
1372 × 4.3 = 5899

Now, Hawk of the Fog grabs some equipment:
● Monet (T): +15 Total DMG, +30% Lightning DMG;
● Ishikawa Goemon (M): +20% Total DMG, +25% Lightning DMG;
● Ishikawa Goemon (B): +35% Lightning DMG;
● 2-piece Ishikawa Goemon: +30% Total DMG;
● Nuada's Revenge: +40% Lightning DMG;
● passive bonuses: +30% Lightning and extra +45% Lightning DMG;
● she is PSY-type and will be attacking a MECH-type enemy: +30% DMG.

Elemental damage multiplication formula (built-in bonuses, stigmatas, weapon, type advantage/disadvantage):
(1 + Total DMG) × (1 + Elemental DMG) × (1 + type advantage/disadvantage or ±0.3)
↓
(1 + 0.15 + 0.2 + 0.3) × (1 + 0.3 + 0.25 + 0.35 + 0.4 + 0.3 + 0.45) × (1 + 0.3) = 6.54

Finding out how much damage Hawk of the Fog will deal when fully geared:
Base DMG × final damage multiplier
↓
5899 × 6.54 = 38579

(1 + Physical/Elemental DMG taken) × (1 + Total DMG taken) is a simple formula for stigmatas, weapons and skills that make the enemy take more damage, like Isaac Newton (B), Mei: Beach Party (M), Blood Dance.
Take note: damage taken from host (Swallowtail Phantasm's Scarlet Chain skill, for example) works just like the aforementioned, but only for the host.
Overall, damage taken is a huge multiplier.

Final damage multiplication formula:
Base DMG × final Physical/Elemental DMG multiplier × damage taken × damage taken from host × stage DMG multiplier (if present)

Following the formula, with applied Newton (B) and Mei: Beach Party (M) + stage effect (enemies take 50% more Physical DMG), Void Drifter's final damage will be:
3500 × 17.54 × (1 + 0.31 + 0.15) × (1 + 0.5) = 134444

Hawk of the Fog's final damage with applied Newton (B) and Mei: Beach Party (M) + stage effect (enemies take 50% more Elemental DMG):
5899 × 6.54 × (1 + 0.31 + 0.15) × (1 + 0.5) = 84489

Planck (M) provides 51% Physical DMG.
Marco Polo (M) provides 50% Crit DMG.

Planck (M)
50% Total DMG × (50% Physical DMG + 51% Physical DMG) × base 200% Crit DMG
1000 base ATK × [final multiplication value]
↓
(1 + 0.5) × (1 + 0.5 + 0.51) × 2 = 6.03
1000 × 6.03 = 6030

Marco Polo (M)
50% Total DMG × 50% Physical DMG × (base 200% Crit DMG + 50% Crit DMG)
1000 base ATK × [final multiplication value]
↓
(1 + 0.5) × (1 + 0.5) × (2 + 0.5) = 5.625
1000 × 5.625 = 5625

6030 > 5625
Planck (M) > Marco Polo (M)

Monet (T) provides 15% Total DMG, 30% Lightning DMG and 117 ATK.
Shakespeare (M) provides 51% Lightning DMG and 91 ATK.

Monet (T)
(50% Total DMG + 15% Total DMG) × (50% Lightning DMG + 30% Lightning DMG)
(1000 base ATK + 117 ATK) × [final multiplication value]
↓
(1 + 0.5 + 0.15) × (1 + 0.5 + 0.3) = 2.97
1117 × 2.97 = 3317

Shakespeare (T)
50% Total DMG × (50% Lightning DMG + 51% Lightning DMG)
(1000 base ATK + 91 ATK) × [final multiplication value]
↓
(1 + 0.5) × (1 + 0.5 + 0.51) = 3.015
1091 × 3.015 = 3289

3317 > 3289
Monet (T) > Shakespeare (T)

Thales (M) provides 20% Total DMG and makes the enemy take 20% more Fire DMG.
Shakespeare (M) provides 51% Fire DMG.

Thales (M)
(50% Total DMG + 20% Total DMG) × 50% Fire DMG × 20% Fire DMG taken
1000 base ATK × [final multiplication value]
↓
(1 + 0.5 + 0.2) × (1 + 0.5) × (1 + 0.2) = 3.06
1000 × 3.06 = 3060

Shakespeare (M)
50% Total DMG × (50% Fire DMG + 51% Fire DMG)
1000 base ATK × [final multiplication value]
↓
(1 + 0.5) × (1 + 0.5 + 0.51) = 3.015
1000 × 3.015 = 3015

3060 > 3015
Thales (M) > Shakespeare (M)

Thales (M) provides 20% Total DMG and makes the enemy take 20% more Fire DMG.
Leeuwenhoek (M) provides 20% Total DMG and 20% Fire DMG.

Thales (M)
(50% Total DMG + 20% Total DMG) × 50% Fire DMG × 20% Fire DMG taken
1000 base ATK × [final multiplication value]
↓
(1 + 0.5 + 0.2) × (1 + 0.5) × (1 + 0.2) = 3.06
1000 × 3.06 = 3060

Leeuwenhoek (M)
(50% Total DMG + 20% Total DMG) × (50% Fire DMG + 20% Fire DMG)
1000 base ATK × [final multiplication value]
↓
(1 + 0.5 + 0.2) × (1 + 0.5 + 0.2) = 2.89
1000 × 2.89 = 2890

3060 > 2890
Thales (M) > Leeuwenhoek (M)

Impair reduces the enemy's resistance to physical damage by lowering its DEF value.
The DEF calculation formula:
1 ÷ (1 + (enemy's DEF value ÷ 300 + 200 × valkyrie's level))

Now, simple example:
● the enemy has DEF value of 996 (most of Exalted Memorial Arena bosses have 996 DEF);
● the enemy is being attacked by a level 80 valkyrie, therefore her DEF effect value is 1900.
The result:
1900 ÷ (1900 + 996) = 656 or 65.6% which is the final multiplier. 34.4% of physical damage is lost because of the enemy's DEF.

Now let us multiply Void Drifter's final damage (134444) by this value.
134444 × 0.656 = 88195 which is the final damage.

Let us use Celestial Hymn's or Herrscher of Sentience's ultimate skill to reduce the enemy's 996 DEF by 75%:
1900 ÷ (1900 + ((1 - 0.75) × 996)) = 0.884

Multiplying Void Drifter's damage:
134444 × 0.884 = 118848
Thus, (88195 ÷ 118848) × 100 = 100 - 74.2 = 25.8% increase in damage, and the difference will be even bigger with bigger numbers.

The data used in calculations was taken from this data sheet[docs.google.com]

Important:
● the Impair effect gets replaced by a new one every time you apply it (if your Celestial Hymn has applied a 75% Impair with her ultimate skill, Divine Prayer's Divine Penalty passive skill (50% Impair) will replace the first Impair).

Some enemies have innate physical resistance which is different from DEF and shields.
I will borrow some information from SONA's guide[www.hoyolab.com] on HoYoLab: «Base physical resistance stacks multiplicatively with DEF. Exalted Memorial Arena Benares has 996 DEF. When not impaired, Benares has (1900 ÷ (1900 + 996)) × 0.5 = 67.2% physical resistance».
Total physical resistance can be calculated as:
Total physical resistance = 1 ÷ (1 + (DEF ÷ ((300 + 20) × valkyrie's level)))

Physical breach directly reduces an enemy’s physical resistance.
Physical breach damage increase can be calculated as:
Physical breach ÷ ((1 - (1900 ÷ (1900 + DEF))) × innate physical resistance)

Elemental Breach is a way of countering the enemy's elemental resistance. Currently only Azure Empyrea and elemental Herrschers have this ability of reducing the enemy's elemental resistance.
Let's ask Azure Empyrea to make an example. She has a 14% Elemental Breach (at S-rank).
If an enemy has a 50% elemental resistance, Azure Empyre shall decrease it to 36%.
Now, let's say, thou can deal 50000 Ice DMG, alright. The difference in damage output:
● without breach: 50000 × (1 - 0.5) = 25000;
● with breach: 50000 × (1 - (0.5 - 0.14)) = 32000.

The difference seems insignificant to thee? Watch this, then:
● without breach: 1000000 × (1 - 0.5) = 500000;
● with breach: 1000000 × (1 - (0.5 - 0.14)) = 640000.
The difference between 500000 and 640000 is 140000. Quite a lot, is it not?

But these are just beautiful numbers. Let's take more realistic ones, now.
Remember how much damage shall fully geared Hawk of the Fog deal with Kei: Taranic Talons combo attack? If not, 38579.
If the enemy has a 50% elemental resistance, then:
● without breach: 38579 × (1 - 0.5) = 19289;
● with breach: 38579 × (1 - (0.5 - 0.14)) = 24690.

The more elemental resistance the enemy has, the more valuable Elemental breach gets.

Important:
● most of Exalted Memorial arena bosses have 31% elemental resistance;
● Elemental breach can stack from different sources.

If you are a Genshin Impact player, then these may look familiar to you, but anyway, affixes are randomly rolled extra bonuses that a stigmata can provide.
Each stigmata can have up to two slots and you can change the affixes, if you got the ones you do not need, or try to roll for better ones, with bigger numbers.

To roll for affixes, you need these thingies here:

While using these, if a stigmata you are trying to make better has only one slot available, there is a chance of getting an extra slot for free.
Quite easy to obtain, especially if you have a lot of spare 4★ trash stocked up in your inventory.

No luck on getting an extra slot? Use these:

Also easy to obtain. Just do not miss on the open world.

Got a good affixes, but the second one is not that good? Lock the good one, then roll again!

These are the hardest to obtain and more accessible to level 81+ players. If you are under level 81, try to get these from Elysia's Realm. Level 81+ players do not need to be said where to get these, right?

Always aim for ATK. It is the most important and beneficial both for physical and elemental damage valkyries. You should know by now that ATK is very important because it gets multiplied several times resulting in many times bigger numbers of final damage.

You may get these instead of just ATK:
MECH-type characters gain XX ATK
BIO-type characters gain XX ATK
PSY-type characters gain XX ATK
QUA-type characters gain XX ATK
IMG-type characters gain XX ATK
Dual pistols and lance wielders gain XX ATK
Katana and cross wielders gain XX ATK
Cannon and scythe wielders gain XX ATK
Greatsword and gauntlet wielders gain XX ATK
Archers and Chakram wielders gain XX ATK
These are completely alright. It is the same ATK, but only for a certain type or weapon wielders, which means, let's say, MECH-type characters gain XX ATK will not give that ATK to a BIO-type valkyrie (obviously).

Just to clarify:
● Kiana Kaslana, Kallen Kaslana and Eden are dual pistols wielders;
● Raiden Mei and Yae Sakura are katana wielders;
● Bronya Zaychik is a cannon wielder;
● Murata Himeko, Liliya, and Rozaliya are greatsword wielders;
● Theresa Apocalypse, Mobius and Griseo are cross wielders;
● Fu Hua and Carole are gauntlet wielders;
● Seele Vollerei and Rita Rosweisse are scythe wielders;
● Bianka Ataegina (Durandal) and Asuka are lance wielders;
● Amy (Fischl), Elysia, and Natasha «Raven» Cioara are archers;
● Pedofallus and Aponia are chakram wielders.

Crit DMG and CRI affixes are acceptable for physical damage, but overall ATK is more important. CRI is CRT which is used in finding crit rate. If you did not already read it, Enduhchi recommends checking their Honkai Impact 3 basics: how to find a valkyrie's crit rate guide.

Attack deals XX more Ice DMG
Attack deals XX more Fire DMG
Attack deals XX more Lightning DMG
Not what you could have thought. These do not increase elemental damage. Instead, they make thy valkyrie's attacks deal a set amount of extra damage. Give a stigmata with fire damage affix to Goushinnso Memento (she deals ice damage), and she will deal a tiny bit of fire damage along with ice damage. Give her an ice damage affix, and she will deal the same amount of ice damage + extra tiny bit of ice damage. Not that big of a deal.
Might be used in some cases. For example, a Memorial arena boss, Shadow Knight, which takes more physical damage after taking some elemental damage.

Recovers XX SP/s when enemies are nearby
As its description says, stay close to an enemy, and you will be getting a certain amount of SP per second. Might be useful when you have issues with getting the ultimate skill ready.

Recovers XX HP/s when enemies are nearby
Staying close to an enemy heals you. Kind of… Might be useful for Herrscher of Flamescion to help her slowly regenerate her shield.

Gain XX HP
Increases a valkyrie's max HP. Err, forget about this and move on.

Gain XX DEF
Might be useful for Yamabuki Armo… Aw, come on, mate. No one needs it.

Please give this guide a thumb up, so it will be more popular and more players will easily find it.


Related guides:
Attack types and game mechanics
How to find crit rate
5★ stigmatas
Fu Hua