I got a youtube channel with a couple helpful guide you may want to take a look at. While I never covered how all the weapon types work against each other, I think I could give you a couple pointers.
Basically you want to use psionics attacks against units with high armor, and/or little to no shields. Psionic attacks bypass (or ignore) armor, but not shields. So you can use units like the the syndicate overseer to take out quartzites and other units that have loads of armor and no shields, like the dvar units.
Likewise, you want to use melee attacks (all damage channel work pretty much the same way with some units have a a bit more resistance to some damage channels; ex: assembly get arc damage resistance) against units with high shields and little to no armor. Melee attacks bypass (or ignore) shields. So you don't really want to melee attack targets with high armor, but you do want to use them against targets like the the psifish, who, without mods, have no armor. Of course closing the distance with psi-fish is tricky, unless you can get high shields on your units through mods or abilities like swarm shielding (that is the thing the kir'ko get when they stand next to each other).
Damage channel resistances work like having more armor or shields, but only when they resistance matches the incoming damage.
The equation for damage taken is:
- Damage resistance = sum of shields and elemental resistance or armor and elemental resistance
So if you attack a target with 0 armor, 0 shield and 0 arc resistance with a melee attack that does 10 arc damage, then it takes 10* .9^(0+0+0)= 10* 1= 10
If you attack a target with 0 armor, 0 shield and 2 arc resistance with a melee attack that does 10 arc damage, then it takes 10* .9^(0+0+2)= 10* .9 *.9= 8.1
If you attack a target with 1 armor, 0 shield and 0 arc resistance with a melee attack that does 10 arc damage, then it takes 10* .9^(1+0+0)= 10* .9= 9
If you attack a target with 0 armor, 1 shield and 0 psi resistance with a psionic attack that does 10 psionic damage, then it takes 10* .9^(0+1+0)= 10* .9= 9
If you attack a target with 1 armor, 0 shield and 0 psi resistance with a psionic attack that does 10 psionic damage, then it takes 10* .9^(0+0+0)= 10* 1= 10
I also found this info on the wiki:
Arc
- Anti-Mechanical
- Bouncing Projectile
- Stagger
Biochemical
- Armor Melt
- Infections
- Slowdown
Kinetic
- Anti-Organic
- Bleeding
- Smart Bullets
Thermal
- Blinding
- Burning
- Disintegration
- Shield Breaking
Psionic
- Buffs
- Debuffs
- Mind Control
- Ignores Armor
Hope this helps!