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 Centers. The 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 medical schools, 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 individuals in specified fields who are pursuing an advanced degree in palliative care or related fields, and (4) 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.