There's no specific co-op game mode, you just need to start a game as normal and then have Player #2 press "START" or "A" on their controller to join the game. From that point on, you're on a co-op game.
Your shields are shared! If the team has zero shields and one of you gets hit then you get knocked down and a death timer of a few seconds starts ticking. The surviving player can rescue the downed player by simply standing close to them for a second or so, after which you'll both be active again but still on zero shields.
If while one player is knocked down their death counter hits zero or if the other player gets hit themselves before the rescue can be completed, then it's game over.