Whan using weapons with different damage types
Normal, Elemental, Piercing and Shadow.
How much damage is added if the monster is weak to that attack?
How much damage is lost if the monster is strong to that attack?
Is there any hard numbers on this?
You can enable damage numbers in Options in-game. It won't show you the monster's total HP, but the numbers will be gray if the monster is resistant to your damage type or will be bolded and orange if the monster is weak to it.
Please see this Wiki page for more:
http://wiki.spiralknights.com/Damage