Forget about playing polo and just start swimming to build up your endurance and your speed, that's all the matters... the better swimmer will win.
as far as the egg-beat kick. The way I learned is imagine that there are walls on the sides of your feet, start by just simply "kicking the walls" one leg at a time kick to one side "like you're kicking down a wall" then the other, eventually you'll get comfortable with this and start twisting your ankle to create the egg-beat effect.