Firstly, you didn't ask, but movement is effectively cut in half, rather than quartered. Per the rules on climbing, swimming, and crawling:
Each foot of movement costs 1 extra foot (2 extra feet in difficult
terrain) when you’re climbing, swimming, or crawling. You ignore this
extra cost if you have a climbing speed and use it to climb or a
swimming speed and use it to swim.
So movement is ⅓ if the heroes are in difficult terrain - normal swimming is only ½.
Next, spellcasting. There are no specific rules about underwater spellcasting, but the description of verbal components says:
Most spells require the chanting of mystic words. The words themselves
aren't the source of the spell's power; rather, the particular
combination of sounds, with specific pitch and resonance, sets the
threads of magic in motion. Thus, a character who is gagged or in an
area of silence, such as one created by the silence spell, can't
cast a spell with a verbal component.
It's pretty reasonable to extend this to a character underwater, so spells with verbal components are out. There doesn't seem to be anything else that would restrict underwater spellcasting, however.
Finally, breathing mechanics. The rules for suffocating are on page 183 of the PHB, and are much simpler:
A creature can hold its breath for a number of minutes equal to 1 +
its Constitution modifier (minimum of 30 seconds).
When a creature runs out of breath or is choking, it can survive for a
number of rounds equal to its Constitution modifier (minimum of 1
round). At the start of its next turn, it drops to 0 hit points and is
dying, and it can't regain hit points or be stabilized until it can
For example, a creature with a Constitution of 14 can hold its breath
for 3 minutes. If it starts suffocating, it has 2 rounds to reach air
before it drops to 0 hit points.