Preparing for Battle
Before fighting any boss in the game, find the nearest red and green crystals and refill your health and elemental power. You might have to wait around until they keep regenerating, but you will not last long in battle unless you are at full capacity for both.
Fighting Elementals
If you are attacking a boss and strangely not gaining any hits, it is an elemental boss. The dragons have one specific power element each that is its weakness, and any other elements will be ineffective. Do not use up all your time or power sources trying to fight the boss with the wrong element. Strike once with each until you find out which one actually works. You will know you have the right element when the boss's health starts to go down. Move around while you test each one out to make it more difficult for the elemental boss to strike you down or hurt you.
Preventing Damage
A large amount of the game is played in a volcano-like area, and there are pits that spew fire and hurt you. Walk, don't fly, in these general areas when you're in co-op mode. The other player may accidentally pull you right into the fire since you are bound together, and you don't have as much control or as fast of a reaction if you're in flight.