If you get the exposure right you should not have to do a lot of post processing on moon shots. In other words they should look just fine right out of the camera. If they don't look just fine, and you have to do lots of post processing, then that is a sign that you have the exposure wrong.
Since you'll typically be shooting at a faster shutter speed you don't really need mirror lockup or a cable release. The moon is being lit by direct sunlight, so that means you would expose it about the same as any other subject lit by direct sunlight. I've shot lots of moon shots handheld that came out perfectly. I would typically shoot these at about 5.6 at 1/250th or 1/500th at ISO 100 using my Sigma 70-200 at 200mm.
One large factor, however, is going to be the amount of haze or clouds. Sometimes there will be light clouds that you can't really even see or haze that you can't see, but this will affect the exposure. So what I generally do is to shoot 6 shots at 5.6 or 8 and vary the shutter speed for each. I then take a look at them on the computer and select just the moon and look at the histogram in PS. That way you can pick the proper exposure for the conditions exactly. I shoot moon shots from time to time to just get different phases and different colors (made possible by the amount of haze, dust in the air, etc. Also I get some with light clouds, some while it is still daylight, etc. I keep them in a special folder and then when I need a moonlit scene, I just add the appropriate moon to the shot using PS.