Hi "Starbie"! Welcome to the wonderful world of observational astronomy. Yes, the Moon does move very quickly across the sky. But not as fast as the stars. If you pay attention (and from your post, you certainly are), you will find that the Moon "drops back" one angular diameter of itself an hour. If you go out on your deck some night and watch the Moon near a first magnitude or some other visible star, you will see this.
Enjoy the Moon! And clear skies always to you, my friend. 🙂