What Do Elements Really Do?

Salted already explained how earth was slow and hard hitting, air was high speed low damage, fire tanking and slow speed, thunder was fast hitting and crits and water was better spells but weaker melee.

I still don't know however if these are attributes associated with the weapons and armors focused on certain elements or if upgrading strength actually reduces attack speed.

Could someone explain?

 

Upgrading strength will not lower attack speed, but, as you said, most items associated with earth that would benefit from a strength upgrade are also associated with slow attack speeds. What the elements themselves do is pretty basics it's just a special type of attack damage.