I'm in the Romulan tier of missions right now, and I keep losing to the Klingon fleets. I'm playing the missions set on advanced. I notice that they seem to be hitting me with something that does DoT and it hits my shields from all sides. Also, I notice them damaging my ship even when I'm at full shields. How can I counter them?
Armor in space combat (as presented in this game) is pointless. Armor only reduces damage (provides resistance) once the attack has penetrated your shields. You're better off using a console slot to increase your shields, shield / hull repair skills, or shield regeneration abilities (lessening how much damage reaches your hull in the first place AND increasing your ability to repair all damage (not just damage of a specific type)).
Repair skills / abilities are plentiful, and if you instead maximize those, you'll be able to have one active at all times. These work against all damage types, where-as armor only provides resistance against certain types of damage.
D&D DM/Player since 1982 - all versions except the despised 4e
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As for being hit... You probably need armor... You can see a list here, to help you match your current layout to enemies:
http://www.stowiki.org/Category:Ship_Engineering_Consoles
Repair skills / abilities are plentiful, and if you instead maximize those, you'll be able to have one active at all times. These work against all damage types, where-as armor only provides resistance against certain types of damage.