Well my suggestion would be something like this:
HEAVY INFANTRY
Spearman: Bonus against both cavalry.
Role: Basic unit available to every civilization. Main unit and useful to protect other units from cavalry. Meatshield
Swordsman: Higher base attack than spearman, bonus against heavy infantry.
Role: Break melee infantry fronts, kill unprotected units (light infantry, support or siege)
Pikeman: High HP and Armor, low attack (if no formations are implemented, every unit gets a bonus attack aura for other pikes). Bonus against cavalry. Slow movement.
Role: Protect ranged infantry, capture or when massed act like a powerful slow spearhead.
-LIGHT INFANTRY:
Skirmisher: Shortest range. Higher base attack
Role: Basic ranged unit. Useful against everything, but with low range it's the most vulnerable unit. The best ranged unit to deal against both cavalry types
Slinger: Medium range. Bonus against heavy infantry
Role: Better than skirmishers against heavy infantry. Should win skirmishers by range, but they damage between them could be on pair. Archer: Long range. Low attack and bonus against Light Infantry.
Role: Kill skirmishers and slingers and support long range attack. All light infantry get a bonus against Light Cavalry.
-HEAVY CAVALRY
Sword cav: Fast. Bonus against light infantry
Role: Kill light infantry (altough spear cav should be better), raid, and maybe catch light cavalry
Spear cavalry: Tankier armour, HP, Good attack. Slower
Role: Better frontline cavalry than swordman cavalry.
-LIGHT CAVALRY
Skirmisher cavalry: Good damage. Low range
Role: Hit and Run, effective against everything, but could be killed if get caught or by ranged units (so it would be mainly anti heavy infantry)
Archer cavalry: Low damage, Long range. Bonus against light units (both infantry and light cavalry). Less HP than skirmisher cavalry
Role: Hit and Run from distance, but worst DPS against heavy. Loose against ranged infantry if gets caught in range (specially against archer, with same range)
All defensive buildings get bonus against Light Cavalry
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Then, I would make Spearman and Skirmisher infantry and Skirmisher cavalry costing only food and wood (Trash unit). More uncommon units could cost food/wood and metal (no stone for siege or slingers, only for buildings).
I would change hacking/pierce/crush to Melee/Ranged/Siege. (I think it could be easier to balance and to think, although hybrid damage could be special feature for some units)
*I know that there's some historical things... some slinger could out range archers, and the spear cavalry role resembles more a medieval knight-like... but I think that is somewhat cohesive system.