My priority for weapons upgrades is:

Knights / cavalry
Halbediers / axemen
Swordsmen/melee infantry
Spears
Missile units

I give my mounted units weapons upgrades because they’re best bet in a fight is to kill enough enemy on the charge that they route – once they get caught in a melee they just start to die. I upgrade my halb’s, axemen, etc. for similar reasons – they have poor moral and need to kill a lot of men quickly before they route or before the enemy knights can break off for another charge. You can get a lot of use out of cheap armor-piercing units like urban militia, vikings and woodsmen with just a few upgrades that you only have to pay for once. Swordsmen need the upgrades, but I make them wait as they can generally already hold their own against most other melee and sword units.