They have a few different weapons at their disposal, most of which you won't see unless you damage them. First, they release swarms of "thargons" that are basically tiny fighters that can fire, and act as homing missiles. ECM and fighters can distract the thargons during a charge. The Thargoid itself has a basic weapon that bleeds through shields, but your shield will absorb most of the damage. The projectiles are red, and slow firing. Makes a 'shew shew' kinda noise when firing. It's hard to describe.
If you manage to destroy one of its hearts, it will begin using what basically looks like a yellow lightning attack. I you're within 700 meters, it will attempt to use this every now and again. It will mess up your shields, and partially disable your ship, slowing you, and making it harder to move. Basically like a snare attack from an RPG.
If you manage to take out multiple hearts, it will start to use caustic missiles, and the classic shutdown pulse. The caustic missiles are scary, though, as you cannot avoid them. (ECM has no effect.) They cause constant hull damage over time. Lastly, if you destroy a Thargoid somehow, it will leave behind a caustic cloud. Do not enter it, unless you have a way to repair, (station nearby, or repair limpits) as I believe once that gets on your hull, it won't stop eating away your ship until you get to a station.