Standard Skill
Tapping Dawnwing's Standard Skill causes her to dash in the direction she is moving in (or forward if she isn't moving) and create 1 Crystal Vane.
If you hold the skill down it will cause Katya to do two dashes, and create 3 Crystal Vanes. (whilst still using the same S-Energy as a single dash)
The two dashes don't have to be in the same direction - you can dash one way, dash the opposite way, and end up back where you started.
This is recommended and doesn't require much practice to do.
Further note that the dash (as well as Dawnwing's regular dodge) is more of a phase, and does allow her to go through most enemy-created barriers.
That Dawnwing's standard Skill moves her so much is what gives Dawnwing so much mobility.
Dawnwing can have a max of 12 Crystal Vanes at once.
Dawnwing's Standard Skill does also deal some PBAoE Frost damage when creating a Crystal Vane.
This isn't much damage, but it's something, and is good enough for clearing trash mobs.
Against bosses, staying near the boss so this PBAoE damage hits them will increase your damage a little.
You may wish to get Dawnwing to 12 Crystal Vanes before using a debuff (e.g. Celestial Swan), so that you don't need to create Crystal Vanes during the debuff and can just do damage to make the best use of the debuff as you can.
Note that some in-game effects treat the two parts of Dawnwing's Standard Skill dash as separate skills/effects
For example the effect "50% of S-Energy consumed when using a skill is converted into U-Energy" buff gives U-Energy for both the first and second dash
(Granting 4 and 3 U-Energy respectively) [If you only do one dash you only get the first lot of 4 U-Energy]
Passive Effect
Dawnwing has a gauge near the middle of the screen that fills.
This fills slowly when in combat, or quickly when shooting.
(Max is 100. Gain 10/s naturally, as well as 4 per shot hit.)
When full, the next shot Dawnwing fires will reset the gauge and trigger her Crystal Vanes - Up to 6 Crystal Vanes will fire at the shot enemy, dealing high Frost damage.
After proc'ing the passive, some Crystal Vanes disappear (how many disappear depend on how many you have. With 6 or more Vanes up, 3 Vanes will disappear)
Since a maximum of 6 vanes proc, having more than 6 vanes up doesn't increase Dawnwing's damage.
You can use periods of downtime (e.g. enemy invulnerability phases) to build vanes, so that you don't need to make them during combat and can focus on just on proc'ing the crystals rather than making them, because the crystal proc is where most of Dawnwing's damag comes from.
For each Crystal Vane that fires at an enemy, Dawnwing recovers 4 bullets.
With the max of 6 Crystal Vanes firing, this means Dawnwing recovers 4*6=24 bullets per proc.
Due to Dawnwings natural aurora regneration (10/s) means it actually takes her a little less than 24 bullets to proc her vanes, which recover 24 bullets.
This means that [provided you land your shots] Dawnwing never needs to manually reload.
Shooting an enemy builds the gauge, even if the enemy is invulnerable.
This allows you to build the gauge to full whilst an enemy is invulnerable so that when they become vulnerable again your next shot proc's Dawnwing's Crystal Vanes.
Note that in practice you don't need to worry about her Crystal Vanes - just use her standard skill after the vanes fire, and you'll always stay at or above 6.
Dawnwing will also use her standard skill automatically when switched in (This doesn't use any S-Energy or skill charges though)
This does create Crystal Vanes, but since it's her standard skill it does also cause her to move, which can make it a liability in certain situations.
Especially if you only move once and don't dash twice to end up back where you started.
Note that for Dawnwing to double-dash when switched in you need to hold her standard skill key (Default: E) down, not her swap-in key.
This does require some practice.
Ultimate Skill
Dawnwing's Ultimate Skill causes her to shoot a huge arrow at the target that deals Frost damage.
This also causes Dawnwing to use up to 6 Crystal Vanes (if you have them) to fire a ray at the target.
...this does mean that Dawnwing's Ultimate skill is basically her Standard Skill with an extra Frost Damage projectile on top.
Dawnwing's Ultimate Skill also restores all her standard skill charges, allowing you to immediately use your Standard Skill again.
This isn't actually needed with perfect play, but it is useful if you miss some shots and/or are running out of S-Energy due to using your Standard Skill to clear trash mobs.
In Single-Player content, time slows down when you use your Ultimate.
In Multi-player content time does not slow down, meaning that if you haven't disabled ultimate animations then Dawnwing's Ultimates are a DPS loss in multiplayer - it just takes too much time to activate for its damage - you're better off just using the standard damage rotation above (with her Standard Skill) instead.
You can disable ultimate skill animations in the options menu, and if you use Dawnwing in multiplayer content that is something you will very much want to do.
If you don't do that then again - I am not personally convinced her Ult is ever worth using in multiplayer - I think you're better off using her standard skill as normal instead.
The one exception is if you can pre-cast her ult, which can be tricky to pull off.