Although any unit will eventually fall to an overwhelming force of stupid, it is significantly better for your economy to use the right units.

Some counters are due to particular units not having a attack that targets a unit tier, or a weaker weapon that targets lower tiers.

Example: While Maulers have a very strong weapon that can cut through cruisers, that weapon does not target frigates, however its much weaker secondary weapons can. A Mauler would win against an archer, but a couple of archers and a pack of brutes to get in the way which are much cheaper (and don't cost radioactives) would cut through the Mauler.

An important recommendation is "keep the army diverse". Spamming a single unit is almost never a good idea, and leaves you exposed.

If you already have the right counter on the field in your army, it is much better than watching your army get wiped out then waiting a couple of minutes to build the appropriate counters and try to recover.

Below is a rough guide, and is probably full of errors. Feel free to throw a pie in my face if there are some stupid mistakes.

Post-Human Coalition

Hermes
Uses: Scout, Spotter
Weak against: Pretty much anything that moves, Zeus, Drone Hive, Avenger, Hades, Furies, Punishers, Dominators

Brute
Uses: Cannon fodder
Weak against: Zeus, Avenger, Drone Hive, Hades, Furies, Punishers, Dominators

Archer
Uses: Light direct DPS
Weak against: Zeus, Avenger, Drone Hive, Hades, Furies, Punishers, Dominators

Medic
Uses: Healing
Weak against: Anything with a weapon, Zeus, Drone Hive, Avenger, Hades, Furies, Punishers, Dominators

Zeus
Uses: AoE, Eliminating T1 swarms
Weak against: Nemeses, Maulers, Hades, Punishers

Artemis
Uses: Artillery
Weak against: Nemeses, Maulers, Hades, Punishers

Nemesis
Uses: Sniper
Weak against: Archers (+Brutes), Reapers/Capacitors/Martys, Hades, Punishers

Apollo
Uses: Anti-Air, Anti-drone
Weak against: Archer (+Brutes), Reapers/Capacitors/Martys, Nemeses, Maulers, Artemis, and many more

Prometheus
Uses: Anti-dreadnought dreadnought
Weak against: Hades, Punishers

Cronus
Uses: Indirect-fire Capital ship, siege
Weak against: Hades, Punishers, Nemeses, Maulers, Prometheus, Savager

Hyperion
Uses: Support
Weak against: Hades, Punishers, Nemeses, Maulers, Prometheus, Savager

Hades
Uses: Bombing runs
Weak against: Apollo, Furies, Sky Cleanser, Dominators

Fury
Uses: Anti-air, harassment
Weak against: Apollo, Sky Cleanser

Pan
Uses: Scout
Weak against: Apollo, Furies, Sky Cleanser, Dominators

Substrate

Reaper
Uses: Light direct DPS
Weak against: Zeus, Avenger, Drone Hive, Hades, Furies, Punishers, Dominators

Martyr:
Uses:
Weak against: Zeus, Avenger, Drone Hive, Hades, Furies, Punishers, Dominators

Sky Cleanser:
Uses: Anti-air, Anti-drone
Weak against: Archer (+Brutes), Reapers/Capacitors/Martys, Nemeses, Maulers, Artemis, and many more

Destructor
Uses: Seige vs buildings
Weak against: Archer (+Brutes), Reapers/Capacitors/Martys, Nemeses, Maulers, Artemis, and many more

Avenger
Uses: Eliminating T1
Weak against: Archers (+Brutes), Reapers/Capacitors/Martys, Hades, Punishers

Drone Hive
Uses: Eliminating T1 Swarms
Weak against: Apollo, Sky Cleanser, Archers (+Brutes), Reapers/Capacitors/Martys, Hades, Punishers

Mauler
Uses: Sniper
Weak against: Archers (+Brutes), Reapers/Capacitors/Martys, Hades, Punishers

Retributor
Uses: Taking peoples lunch money
Weak against: Hades, Punishers, Nemeses, Maulers, Prometheus, Savager

Overmind
Uses: Support
Weak against: Apollo, Sky Cleanser, Hades, Punishers, Nemeses, Maulers, Prometheus, Savager

Savager
Uses: Anti-dreadnought dreadnought
Weak against: Hades, Punishers

Punisher
Uses: Bombing runs
Weak against: Apollo, Furies, Sky Cleanser, Dominators

Dominator
Uses: Anti-air, harassment
Weak against: Apollo, Sky Cleanser

Searcher
Uses: Scout
Weak against: Apollo, Furies, Sky Cleanser, Dominators

You just summed up my situation perfectly. Creating routine 'diverse' armies seems to be limiting: the AI will find a way to come out on top if you haven't managed to crush them early on. I'm trying to pick up some tips from what the better players on YouTube & Twitch do, but apart from Draginol there aren't many who provide commentary on why they are doing what they do.

Effective reconnaissance is something else I'd like to improve in the context of building counters. As a fairly weak player, I find it a challenge to keep tabs on the rest of the map & the enemy while simultaneously getting a solid base /diverse force up with a decent economy. But I think good scouting makes for more dynamic & less robotic game-play.

Thanks for the tip. I usually have a pair of engineers working up a farm of generators near my nexus without fail. What I find though is, although this is fine for unit weapon/defense upgrades, the call-down orbital powers are just totally jammed out by the AI (good for them!). Unless I get strong enough and close enough to knock out their jammers, call-down orbitals just don't come into play. I should say that I mostly play 1v1 on small maps v normal AI. Things might be different on larger maps with more players.

I agree with Moomo, on the smaller maps you don't want to be pumping out Quantum generators all the time, especially with 2 engineers as the Radioactive cost is just too high. I often had one engineer at it much of the time until recently when I worked out why my radioactives were always in such bad shape. I do like to get up to 3 of them if possible though and of course on the large maps you can pretty much have an engineer on them all the time and end up with loads of them.