The basic stroke in ice skating provides a good example of Isaac's 3rd law in action. The ice applies an equal, forward force on you, so you go forward.
Skaters stroke at an angle, so part of the stroke is wasted.
You're pushing forward and to the side. The side push is resisted by the edge of your other blade. Actually, this is exactly why sailboats go forward, not sideways. The wind pushes the boats forward and to the side. The sideways push is resisted by the long keel, but the forward push is relatively unresisted.