I have tiny blood spots under my skin on my legs, arms and chest. They are not very noticeable, and they are scattered (not in clusters). I have noticed slight pain in the areas that they are located. What could it be?

Blood spots. These could be small collections of blood vessels under the skin, cherry angiomas. These are entirely harmless and quite common.
See derm. This can't be diagnosed with the information that is available. A history, physical examination & other studies may be needed to determine cause/causes.