Your family doctor may be able to test you for skin cancer. Family doctors are experts in general medicine, including skin health. They provide services that may include examining skin for signs of skin cancer and even doing simple skin biopsies (minor surgery to remove suspicious skin lesions for laboratory examination).
If your family doctor thinks you need more specialized care, he may refer you to a dermatologist (a doctor who specializes in treating the skin) and work together with the dermatologist to maximize your care. Talk to your own family doctor about what is best for you.