The spots can be calluses due to your fingers or thumbs hitting the piano button or guitar strings. You can avoid this injury by wearing thimbles for guitar and maybe gloves while playing piano.
The red spots could be tiny bruises. Deficiency of Vitamin B12, folic acid, or Vit K could be the cause. The other possibility is petechiae, which are pinpoint-sized red dots under the surface of the skin. They can result from autoimmune disorders, such as lupus or rheumatoid arthritis, bone marrow disorders such as leukemia, inflammation of the veins, as a side effect of certain medications like birth control pills. Please discuss this with your doctor and show him the rash. It is difficult to comment beyond this at this stage.