Palliative Care and Hospice Education and Training Act This bill amends the Public Health Service Act to require the Department of Health and Human Services (HHS) to provide support for Palliative Care and Hospice Education Centers. These centers must improve the training of health professionals in palliative care and establish traineeships for individuals preparing for advanced education nursing degrees, social work degrees, or advanced degrees in physician assistant studies in palliative care. HHS may provide support to schools of medicine, schools of osteopathic medicine, teaching hospitals, and graduate medical education programs for training physicians who plan to teach palliative medicine. HHS must: (1) provide Palliative Medicine and Hospice Academic Career Awards to individuals to promote their career development; (2) support entities that operate a Palliative Care and Hospice Education Center; (3) support advanced practice nurses, social workers, physician assistants, pharmacists, chaplains, or students of psychology pursuing an advanced degree in palliative care or related fields; and (4) award grants to schools of nursing, health care facilities, or programs leading to certification as a nurse assistant to train individuals in providing palliative care. The Agency for Healthcare Research and Quality must provide for a national education and awareness campaign to inform patients, families, and health professionals about the benefits of palliative care. The National Institutes of Health must expand national research programs in palliative care.