Okay, here are my thoughts on the subject:
An X-Com Barracuda is equipped with one gas cannon and one Ajax torpedo launcher.
The Ajax torpedo is a standard homing torpedo for use against hostile ships and submarines. It has a high explosive warhead, similar to torpedos carried on modern submarines, but with better range, speed and damage potential. This
link has some information on the specifications of the Spearfish torpedo, the current torpedo of choice for Royal Navy submarines. The Ajax torpedo launcher has a rack of four torpedos, and they cannot be reloaded until the Barracuda gets back home, unlike with the torpedo launchers on a submarine.
The gas cannon is a scaled up version of the harpoon guns used by X-Com in the early stages of the war. It uses compressed gas to propel a solid metal bolt at an enemy target. Devestating at close range, but it lacks the range and accuracy for long range engagements. The bolt probably has an armour piercing head, much like the arrows used by Welsh and English longbowmen at the battles of Pilleth (where Owain Glyndwr defeated the English) and Agincourt.
There is an option of using DUP torpedoes. The DUP torpedo is larger and more powerful than the Ajax torpedo, but is presumably twice as big because a DUP launcher can only carry two torpedos. The use of depleted uranium in the warhead casing makes this torpedo good at penetrating the armour of a USO hull, but there are political considerations involved with such a weapon. Depleted uranium is still highly radioactive and when it is used for artillery shells in battle, it causes long term environmental damage.
The squad doesn't have grenades yet because we have yet to resolve the problem of throwing grenades underwater. It is probable that we will end up with a grenade launcher or mortar designed for underwater use.
The squad has been using harpoon guns for the first couple of missions. The harpoons inflict armour piercing damage, so the head of the harpoon must be far narrower than that of a standard harpoon, just like with Welsh armour piercing arrows. The squad now has access to handheld torpedo launchers - analogous to rocket propelled grenade launchers, and harpoons with high explosive and phosphorus warheads.
Gauss technology is currently being devbeloped by the scientists. It will effectively be plasma rifles that can work underwater and don't need elerium or zrbite. Zr Zager is almost ready to demonstrate a lab prototype to the Commodore.
Modern assault rifles and the laser rifles developed in the first Alien War don't work underwater. X-Com might decide to buy some after the aliens start attacking ports and holiday resorts, something we should consider.
OTHER SQUAD EQUIPMENT
Liquid breathing technology is mentioned in the Profile of Jake Gaston. The technology was first developed in the 1960s, but it is not yet practical. It is likely that more work will go into developing it if humanity starts mining the sea bed as it allows people to dive much deeper than a standard diving suit can manage. The snag is that the human vocal chords werre not designed to work underwater, so X-Com will need to develop (or buy) some alternative communications equipment.
X-Com has yet to develop medikits and motion scanners that can work underwater. It is likely that they will be based on the same principles as the ones developed in the first Alien war, only waterproof.