Work in progress. I might not add more sections if there's not much interest in the game.
Hitting and destroying enemies with the shot or lock-on will increase your multiplier.
The amount of shots fired by normal shots depends on the amount of lucis you have around your mech. Lucis are the glowing blue orbs around your mech.
Using lock-on makes lucis move away from your mech and attack enemies individually. The more lock-ons you target on a single enemy the faster it will be damaged. If you do a full red lock-on it means all lucis will be used for lock-on attacks. The Astebreed mech has more lucis at its disposal which means more shots and more maximum lock-ons.
Focus shot / tap focus
Your regular forward shot. Actually not very useful apart from destroying projectiles/missiles.
Scatter shot / tap scatter
Spread shot. Point blanking with this causes more damage than the forward shot.
Lock-on cone / hold focus
Locks-on to enemies inside a 20-30 degree cone. Release the button to initiate the attacks. Movement speed is about halved while targeting lock-on. Vaguely similar to soukyugurentai's lock-on system. Good for getting multiple locks on a single enemy and setting up a Wild EX attack.
You can set the direction of the cone to be the direction of your movement or reverse to it when starting a game.
Lock-on circle / hold scatter
Locks-on to enemies inside a circle. Lock-on speed for multiple locks on each enemy is much slower than with the lock-on cone but the targeting area is much larger. Release the button to initiate the attacks. Movement speed is about halved while targeting lock-on.
Targeting a lot of small enemies and/or missiles with this is generally the fastest way to increase your multiplier. The slow multi-lock speed isn't much of a detriment if there's a lot of enemies on screen, and since you might not want to kill medium-size enemies with lock-on for scoring purposes.
Killing enemies with any melee attack will apply your multiplier to the kills and then lower the multiplier.
You can use shot and lock-on simultaneously with melee.
Normal melee / tap melee:
Deals damage quickly to enemies around you and cancels yellow bullets. Using melee slows you down about as much as holding lock-on, worth noting if you need to dodge homing lasers or such. The third attack in the combo stops you to a halt for a split second so use it with caution.
Mashing the melee button makes you invulnerable to any yellow bullet patterns as long as they don't spawn near your hitbox or move at very high speed (st3 Asymptote Est, st4 Portal)
Blade dash / hold melee + direction:
Invulnerable blade dash across the screen. Homes on to enemies. Your mech will go through small enemies and instantly kill them. If the dash ends by destroying a mid-size enemy you can quickly chain another blade dash afterwards. If the dash ends normally there is some recovery time.
Regardless of how many enemies you kill with a single blade dash your multiplier only drops as if you had melee killed a single enemy. So using this properly is really important for scoring, your multiplier will barely drop and you can kill large clusters of popcorn enemies in a single go. If there are mid-sized enemies around you can also chain the move pretty safely.
EX Melee & EX Gauge
The EX gauge is filled by destroying enemy projectiles with shot or melee. It's displayed in yellow above the lifebar.
Using an EX attack empties the whole gauge.
Neutral EX attack / push EX with no lock-ons:
Hits all enemies in a circle around you with large damage and cancels projectiles. Using this makes you invulnerable for about 5 seconds, which very useful for survival purposes. However your EX gauge cannot regenerate until the shield has worn out, making this more of a defensive technique.
Astebreed's neutral EX attack has a secondary explosion that hits the whole screen destroying any popcorn enemies, making it more useful for scoring than the regular version.
Dash EX attack / push EX with multiple enemies locked on:
Deals large damage to all locked-on enemies. Anything excluding large enemies will die instantly. You are invulnerable for the duration of the attack. Lock-ons count even if the lock button hasn't been released. The main uses of this are to instant-kill multiple medium-size enemies and to apply your multiplier to any background enemies.
Wild EX attack / push EX with all lock-ons on a single enemy:
Deals a ton of damage to a single enemy. You are invulnerable for the duration of the attack and it lasts a while. Not very useful for scoring in stages but very much so on boss fights because of the damage output.
Taking damage resets your multiplier to 1. Your multiplier will be halved until you are back in health regeneration state, which is about 10 seconds after taking damage. When you are in regeneration state 'Shield Bonus' is displayed under the multiplier.
Bonus enemy waves
Throughout all stages you'll be awarded with bonus enemies if you clear out waves quickly. These are usually pretty lucrative for score, so being quick is awarded throughout the game.
Boss Fight scoring
There are large time bonuses for destroying bosses quickly so speedkilling is key. Your score multiplier affects the amount of points gained by dealing damage to the boss so not getting hit is also important.
Maintaining the multiplier
'Feint' lock-on / lock-on to enemies but melee them before they are destroyed:
Just hitting enemies with lock-on increases your multiplier, so you can lock-on to enemies while finishing them off with melee or EX attacks to keep your multiplier higher.
Combining Blade dash with lock-on circle:
Using lock-on slows down your normal movement, but you can still use the blade dash to speed around the screen while clearing out enemies. Blade dash kills barely reduce your multiplier while the lock-on circle regenerates it very quickly, so using them in conjunction is a very useful technique. At the same time you'll get a good amount of lock-ons on background enemies for EX attacks.
Efficient melee usage
Use as few melee attacks as you can to clear out enemy waves to prevent your multiplier from dropping. Remember that blade dash only counts as one hit even though it has a large hit area.
Shooting projectiles for multiplier:
You can regenerate the multiplier very quickly by shooting down enemy bombs/missiles with your shot. If your multiplier is low and there's a enemy shooting missiles or bombs nearby it might be worth it to milk the missiles for multiplier instead of going for the kill.
All of these techniques and mechanics can be easily tested in the beginning of Stage 1.
Stage 1 >5mil potential on Hard
At the very start keep locking on to small enemies while meleeing medium size ones as quickly as you can. The faster you kill the medium enemies the more of them spawn.
Make sure to kill the bigger background enemies with dash EX attacks to apply the multiplier to them.
For the fastest kill just keep point blanking it, use neutral EX when it starts firing red lasers.
Use a wild EX attack when it starts firing red lasers in the second phase and finish it off.
Stage 2 >7mil potential on Hard
Midboss: Asymptote Esto (1)
finishing her off with a 16x multiplier awards you with 500,000pts
strategy (hard): Immediately full lock her and use Wild EX attack, then melee her yellow shots point blank and do another one with the same lock. Move slightly above or below her and max out your EX gauge again by meleeing the spreadshot. Keep meleeing her and use neutral EX attack when in danger to avoid losing your multiplier.
You get to fight a lot of bonus enemies for killing the midboss very quickly.
In the asteroid section make sure to get high multipliers on the large asteroids and laser shooting drones. More sets of drones spawn the faster you kill them.
For a faster kill save an EX attack for when its form changes and use it on the boss while it's floating in the background.
Stage 3 >11mil potential on Hard
Hardest stage to execute properly in my opinion. Lots of hard to dodge patterns throughout the stage in addition to difficult melee combos for scoring.
The enemy mechs in the first half do not die from a single blade dash, so don't dash in front of them while they're firing
Midboss 1: Enemy Flagship
The flagship constantly fires missiles at you so you have to keep moving throughout the encounter. There are also cannons shots that explode into a circle of yellow bullets.
Enemy mechs keep spawning until the fight times out. Try to get dash EX attacks in on the mechs and flagship turrets simultaneously. The missiles should easily refill your multiplier while the mechs shoot patterns to reload your EX gauge.
For the mech waves that shoot red spreadshots try to move near the right edge of the screen to avoid the patterns completely.
Midboss 2: Asymptote Esto (2), Indestructible
Boss: Asymptote Esto (3)
Stage 4 >8mil potential on Hard
Stage 5 >14mil potential on Hard
Use blade dash and neutral EX attack a lot to avoid the lucis capture attacks in the beginning. Avoid mashing normal melee or you'll be moving too slowly to dodge them. If all your lucis get captured you'll be unable to get high multipliers in the next section!
Boss: Asymptote Void
Make sure to destroy all parts of the boss. The tail seems to be destroyed if you block the tail swipe with a melee strike after having destroyed all other parts.
Stage 6 >7mil potential on Hard
Milk the small enemies the huge spawners shoot out for multiplier and try to melee as many spawners as you can.
Boss: Queen System
Suddenly you get danmaku patterns, but don't worry as you can cancel 80% of them. Hard to explain strategies for the speedkill in text, you're best off looking at a replay and just practicing the fight. Lots of Wild EX attack usage in any case. Don't mash melee in the final form or you'll move too slowly to dodge the red shots.