Without friction, the best way to move would be with rocket propulsion. If there are solid objects on this frictionless surface, it would also be possible to move by pushing off of them.
Rocket propulsion and pushing off of objects are both possible because of the physical principles described by Newton's Third Law of Motion: Every action has an equal and opposite reaction. For example, using a fire extinguisher for rocket propulsion would send the foam from the extinguisher flying in one direction and the user would be pushed in the opposite direction. On a completely frictionless surface, something as simple as a person blowing air from his mouth would allow for movement. By pushing the air one way, a person or an object is sent in the opposite direction.
The same sort of thing would happen when pushing against an object in the environment. An example of this can be seen in the real world; ice is a very low-friction surface, so if a stationary ice skater pushes another stationary ice skater, they both move in opposite directions.
Walking does not work on a frictionless surface because taking a step requires gripping the ground through friction. The foot usually stays in place while the rest of the body moves forward. On a frictionless surface, a person trying to walk would essentially be doing the moonwalk in place.