Lenalidomide is a drug that interferes with the development of tiny blood vessels that help tumors grow. Lenalidomide in combination with dexamethasone is approved by the Food and Drug Administration (FDA) for the treatment of relapsed multiple myeloma. It is also approved for the treatment of specific types of myelodysplastic syndrome (MDS), another blood cancer. Other research studies using lenalidomide in combination with other drugs in subjects with newly diagnosed multiple myeloma also show good response rate. High dose melphalan is approved by the FDA and is commonly used in multiple myeloma treatment prior to stem cell transplantation. This combination of lenalidomide, high-dose melphalan and stem cell transplantation has not been studied in newly diagnosed and relapsed multiple myeloma, so it is considered experimental. In research studies, "experimental" refers to a drug or procedure that has undergone basic laboratory testing and received approval from the US Food and Drug Administration (FDA) to be tested in human subjects. A drug or procedure may be approved by the FDA for use in one disease or condition, but be considered experimental in other diseases or conditions. In this study, lenalidomide will be given together with melphalan (chemotherapy) with the hope that more disease will be killed before the stem cell transplant. Three months after the transplant, patients will take lenalidomide again with the hope that this will help prolong the time when the disease is in remission.