Noise appears to be more of a problem when image are viewed full-screen on a monitor or when printed large. It's not usually a problem on 4x6 prints, so you need to evaluate the purpose of the image and decide if your worries are unnecessary. Noise doesn't look bad on the shruken image for posting on the web. The EXIF data was stripped from the image you posted, and you didn't post the settings, making it tough to give specific advice.
Stablize the camera by using a tripod. If you don't have a tripod use a table, chair, shelf, or other surface, and fine-tune the camera height with books or any other solid things you can find. Even with a camera stablized on a tripod or table, the act of pushing the shutter button introduces camera shake, so use the camera's timer delay feature. When taking the picture, don't lean on the tripod/table and don't move around; just stand still during the exposure. Set the camera to manual mode, use a low ISO setting (ISO 100), a moderate aperture (f/8), and a slow shutter speed (a few seconds). Review the image. If it's too dark use an even slower shutter speed; if it's too light, use a faster shutter speed. Take another picture, review it, and adjust the shutter speed again, if necessary. Keep doing this until you get the exposure that you want. You might want to turn on other lights in the room to expose the tree a little more. It can be tough to judge an image using the camera's tiny LCD screen, so once you think you have a good exposure I suggest "bracketing" the shot by taking another image or two slightly darker and slightly brighter than what looks good on the LCD. Then review those images on the computer screen, pick the best one and complete any needed post-processing.
