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I have a point and shoot digital camera, with a 26x optical zoom and 10 mega-pixel resolution.
What are the adjustments that I should do to get a nice photograph of moon in the night?
What are the most appropriate settings for shutter speed and aperture? Do I need to set ISO sensitivity to a particular value or let the camera decide it automatically?
I would use the lowest ISO setting available - you don't need much sensitivity - the Moon is a source of light after all. This is also why I would choose a fast shutter. If you can set your camera to different modes, I would start with an aperture priority mode, zoom in as close as it gets using optical zoom (some cameras offer programmatic zoom, but don't use that because your photo will have lower quality).
Your camera will want to measure the light reflected from the Moon, and the black background and average the frame for what it considers a good exposure. If the moon does not fill your frame, the camera will want to overexpose to compensate for the black background. If you fill the frame with the Moon you get a better chance for a good measurement.
Naturally, if your camera lets you control the exposure triange - ISO, shutter and aperture then you can just experiment until you get a good exposure.
Here is some more info about photographing the Moon:
You have got clearly a challenge in front of you, so try to make things work in your favour, reducing the potential issues.
Use a tripod to reduce vibrations. Wait for an adequate night, without too much humidity/fog in the air.
You will not be able to capture a large size picture, for lack of focal length. Then, compose the shot to comprehend other stuff (landscape, details, whatever).
For the iso setting the other answers are already good: the moon is bright so you don't need to go high. Exposing for the moon will probably make all the rest a silhouette.
The moon is lit by the full sun, so its actually as bright as something mid-day.
Sure, it only covers a tiny fraction of the sky, but its bright, especially if you are shooting a full moon.
