I recently bought a new telephoto lens for my Nikon D5000. With the new lens I started taking pictures of the moon.
When I take the pictures I have blue, red, and other colored dots in the photo, seen here: https://flic.kr/p/yxTo43
I do not believe that they are stars.
Does anyone know if they are stars, or something I need to adjust with my camera's settings?
According to your EXIF data (shown on the linked page) you had ISO 3200, which may have introduced noise. Your shutter speed was quite fast, only 1/2500 sec.
I would suggest to lower the ISO and have a longer exposure time. Here is an online exposure calculator for astro photography for avoiding star trails. This photo SE answer explains the background of the calculation (rule of 600).
The Nikon D5000 also has some noise reduction options. Especially interesting for astro photography is the long exposure noise reduction, which may be used for exposures longer than 8 seconds.
When you use long exposure times, you will get some digital noise due to the way the camera works. I am certain that what you see isn't stars.
Imagine a grid of photosensitive spots. Each spot gets light as long as you have the shutter open. Some spots register red, some green, som blue. When reading the light value of the spots, the values of the neighbours are considered in order to get the correct value for all colors at that spot. If for some reason the neighbours have low values, a blue, red og green pixel will appear on your picture. How many depends on the camera and sensor, and the algorithm for calculating the colour values.
To reduce this, you can increase iso slightly, and decrease exposure time. You could also look at getting a lens with a lower f value, which means that wider apertures and more light can come through. The result of doing that is blurring in the edges (which won't matter with moon shots)
