Exactly. Overcoming Earth's gravity well is the reason building a space elevator is such an exciting prospect.

The geosynchronous orbit is about a 10th of the way to the moon, so this moon elevator would go 9/10ths of the way toward the Earth. That's significant but it's precisely that remaining 10th of the way where the vast majority of energy expenditure occurs. If you can make it from the surface of the Earth to geosynchronous, then making it to the moon is relatively cheap.

Rocket to Moon > rocket to orbit > rocket to reach orbit height vertically, so it should help, shouldn't it?

Of course pulling up using the cable would needs power, but it could be a solar powered elevator.

Well, thanks to the bastard rocket equation, fuel savings have an exponential impact. If you can get to geostationary orbit and meet up with the elevator, then fuel up again at the moon, you save a considerable amount of weight in fuel.