Manual. Set it on a tripod. Set it to minimum ISO (probably 100), maximum aperture (smallest number), and play with exposure time until you get something reasonable. Once exposure is good, set a timer so the camera doesn't shake when you push the button. In the event that there is blur from the movement of the moon (I've always used faster lenses, so my exposure time was shorter than yours will be), try cutting exposure in half and doubling ISO (and repeat until you've got a good shot).
Other folks may give different suggestions. This is not necessarily the best way to go, just a way that worked for me. If the results are unsatisfactory, try something else. If there are specific issues, you may need to tweak the process a bit (e.g. if your camera has crappy optics -- as many superzooms do, you may get better shots at a smaller aperture).