Palliative Care and Hospice Education and Training Act This bill requires the Department of Health and Human Services (HHS) to take a series of actions relating to palliative-care training. Specifically, HHS must provide support for Palliative Care and Hospice Education Programs. The programs must support the training of health professionals in palliative and hospice care, including through traineeships or fellowships. In addition, HHS may provide support to medical schools, teaching hospitals, and graduate medical-education programs for training physicians who plan to teach or practice palliative medicine. HHS must also (1) provide Palliative Care and Hospice Academic Career Awards to health professional schools to promote the career development of academic hospice and palliative care specialists, (2) support individuals in specified fields who are pursuing an advanced degree in palliative care or related fields, and (3) award grants to nursing programs 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.