And, we see the moon traveling from east to west in the sky.That is because of the earth's rotation. The moon is actually traveling in the opposite direction, but our rotation makes it appear the opposite.

Also, if you see the moon in a certain spot one night, say at 10pm, on the next night, the moon will be about 12 degrees to the left/east.

Once you start paying attention, you'll notice things about it.

Look up waxing (moon is growing towards full) and waning (moon is getting smaller, towards new).