Maybe this is covered in the DMG, but I wanted to ask other DMs about this. In my last session one of my players casted Firebolt on a Giant Sea Horse (which was underwater), since he rolled high and there's nothing on the creature's stat block granting it resistance or inmunity to fire, the spell worked as normal, but I was left wondering: Would the water provide any sort of cover? Would it mitigate the damage from a FIRE spell? How would it affect a LIGHTNING spell? I know an underwater battle doesn't always happen, but I'm expecting them to dive even deeper into the sea next time, so I wanted to hear some ideas from everyone.

This is one of those things that really depends on how much work you want to do as a DM.  As far as I know there's no official material about this; I think the hand-wavy explanation is that magical fire and magical electricity don't interact the same way with water as their mundane counterparts do. That said, if you want to come up with some mechanics because you think it'd be cool, I agree it'd be cool. Here's some thoughts about what you could do:

Fire: all spells that deal fire damage have their range and area of effect, if any, reduced by half. Half their damage is converted to magical bludgeoning damage, as the superheating of the water creates cavitation bubbles that collapse to harm their targets.

Lightning: Lightning spells are especially difficult to control underwater; all lightning spells gain a secondary effect radius of 10 feet around each target of the spell. Creatures in the radius must make the spell save, taking 1/2 the spell's damage on a failure and no damage on a success. The caster of a lightning spell must always save against secondary damage. Each creature can take only primary or secondary damage from a single spell, and each creature can only take secondary damage once, even if multiple zones overlap.

These are just off the cuff ideas to get the ball rolling. I acknowledge that this could potentially make Lightning spells extremely powerful, but I think the extra danger makes up for that. Obviously this would dramatically effect game balance, so you'd want to adjust your encounters accordingly.

There are rules on underwater combat in the PHB.

In the example you quote a creature fully submerged is resistant to fire damage to RAW the fire bold should have done half damage.

You could rule that something underwater is lightly obscured (this depends on how clear the water is but would be reasonable for a typical ocean on earth).

Anything beyond that is going into homebrew. I would be wary of anything that increases damage. Making a submerged creature vulnerable to lightning damage for example could wipe out a level 5 party. My games are generally pretty light touch on homebrew and the changes we do make tend to be things that crop up often (e.g .you can drink a potion as a bonus action) so I prefer underwater combat as it is written, there are enough changes to make the tactics different than normal combat.

I think environmental effects can be cool and help keep battles fresh, but you don't want to have 15 conditional rules everyone has to remember. I will generally have one thing going on - high winds in a mountain encounter can push at the end of the round, a storm where lightning will strike a metal-bearing character at the end of a round unless they are under cover, etc.

I try to keep the effects relatively simple to understand, flavorful, and try to avoid shutting down anyone entirely. Last time I did an underwater adventure I thought about adding a few additional rules, but decided against it because I just didn't think they added enough value to justify the extra complexity and distraction.