Basically, if you don't have any features that interact with the particular damage type you take, just mark down your hitpoints total and don't worry about the type.
The rules for damage can be found here in the Basic Rules document at DND Beyond. The quotes below are taken from this section. You can also find a pdf download of the Basic Rules here.
Subtract damage from HP.
The rules for damage say:
Whenever a creature takes damage, that damage is subtracted from its hit points. The loss of hit points has no effect on a creature's capabilities until the creature drops to 0 hit points.
So when you take damage, you just subtract that damage from your current hitpoints. Easy enough.
Damage types have no rules of their own.
The rules for damage types say:
Different attacks, damaging spells, and other harmful effects deal different types of damage. Damage types have no rules of their own, but other rules, such as damage resistance, rely on the types.
This means that the damage types are not special on their own. Apart from some feature that respects the damage types differently, fire damage and cold damage will affect the target in the same way - reducing your current hitpoints.
Other features can rely on damage types.
For example, resistance and vulnerability are going to be type-specific:
Some creatures and objects are exceedingly difficult or unusually easy to hurt with certain types of damage.
If a creature or an object has resistance to a damage type, damage of that type is halved against it. If a creature or an object has vulnerability to a damage type, damage of that type is doubled against it.
It is other features that distinguish between the damage types.