I have been experiencing pain and numbing in my hands and wrists (mainly on the right side). The pain usually radiates up to my elbow and my shoulder.

Hand numbness. Many conditions show From the well known as carpal tunnel and pinched nerve in the neck, to circulation problems, vitamin deficiency, diabetes, thyroid disease, neurological disease, stroke , heart problems and the list goes on and . If you have numbness in your hands especially if it is sudden onset and acompanied by what seems to be a serious disturbance in your health = medical atttention.
Wrist and hand pain. This could be carpal tunnel. Do you do a lot of typing or work with your hands? Try a wrist splint for a month and if not better, see an MD for a good exam and workup of cause. If it is not carpal tunnel which is at the wrist level, then it could be at the elbow, shoulder or even neck level.