Barring any freak super-crit from an enemy pet, I can usually take those spirits down with 4 moves from my rare Eternal Strider's Water Pump ability (2 round move). The Strider is a very powerful aquatic pet and difficult to take down because of its self heal. It's like a Mack Truck against any of the Spirit pets, even the Air Spirit which does good damage to Aquatics.
However I really recommend just taking the time to level the Strider to 25 because while it does decent damage at 22-24, the speed at 25 really helps you get the first move that really can win the fights.
Another thing to consider is fight order. I setup my team order and will restart the battle until their order best suits mine. You do not want to be swapping pets and wasting turns in these fights.
I find the hardest fights are the Air Spirit, Earth Spirit, with the Fire Spirit being a little tricky. The Air and Fire Spirit both have secondary pets that do decent team damage. And the Earth Spirit's opening move is a powerful stun. But again, if you look for the best fight order, it's easily to beat these all.