Okay I've been photographing the moon of recent for the "Time Lapse" project (See thread in General Discussions Area), there are a few things that you can do.
First and for most you need to stablise the camera so use a tripod. If you don't have one then you need to come up with an idea of how to keep the camera still by resting it something like a table or bench and elevating the camera so you can see the moon through the eye piece.
Next shutter speed, sounds strange but I assume you are shooting at night, (sometimes the moon can be seen in daylight hours) therefore you will have slow shutter speed which means the camera has to be kept very still. You can try to increase the shutter speed by increasing the ISO this may help a little.
The other thing that I found was metering. In the lense the moon was a relatively small disc in the middle of the large dark background. I switched to Spot Metering so the camera exposure is set by the meter looking at a small area in the middle of the lense instead of metering the whole frame your shooting. Found this increased the exposure from 1/3rd of second to around 1/60th of a second.
Your still going to need the tripod/steady surface to stand the camera on though.
The only other tip I can give is line up the shot and take it using either remote switch that plugs into the camera or use the timer on the camera.
Post a couple of shots and see if we can help you more.
If you want to edit any of my images to demonstrate points/techniques to either myself or others please feel free to do so.
Still learning with Canon 40D, 28-135mm, 100-400mm L and Elements 6.
Any Advice greatfully received.