I'm a big fan of shooting the moon when it's around the new moon phase. The crescent phases a day or two before and after new moon are the best days to view earthshine, reflected light from Earth hitting the night side of the moon. With long exposures, you can pick up the same basic details you would get with a full moon (although the sun-lit parts will be overexposed). The best part is that you can get the nearby stars in the exposure.
I use a SkyWatcher Star-Adventurer tracking mount in order to get the long exposures. If I didn't have this hardware, you would get some movement from the moon/stars in exposures over 2 seconds at this focal length (600mm). This mount combats the Earth's rotation by slowly turning east to west in order for you to stay locked in on one point in the sky. It has a setting for the moon's speed as well, which ends up being important if you're taking pics for longer than a few minutes. I was shooting for about 40 minutes last night and within the photos you can see the moon transit past a few stars as it "slowly" moves west to east.