What are the rules, if any, regarding underwater spellcasting? I have looked through the books and couldn't find anything on the subject. Am I just missing something?
You're missing the 1e DMG.

Seriously, though, if they have water breathing, verbal components should be OK, if you don't pay much attention to material components, continue not to - if you do, well, a lot of them, like IDK, "a pinch of dust" might be problematic when immersed.
A simple rule of thumb might be fire spells do 1/2 damage (a fire based cantrip might just fail, it's just a cantrip afterall) and create regions of superheated water or bubbles of steam that rise rapidly to the surface, lightning spells become spheres centered on the targeted point. Cold spells have reduced area but create ice flows or slush that, again, rises to the surface...spells that produce gas produce bubbles of gas that... yeah, rise rapidly to the surface. ::shrug::
Alternately, you can have spells 'adapt' so they remain functionally the same, but fit the environment. So a spell that conjures a cloud of gas produces it dissolved in the water, working about the same if you have gills, one that produces a cloud of obscuring smoke produces a cloud of obscuring ink, squid-style, a fireball creates a ball of boiling water that rapidly dissipates doing the usual amount of 'fire' (scalding) damage, a Gust of Wind (is that even still around?) creates a sudden current, etc...