Beneficiary Education Tools, Telehealth, and Extenders Reauthorization Act of 2019 or the BETTER Act of 2019 This bill extends funding for, and makes a series of changes to, Medicare enrollment, payment, and prescription drug programs. Among other things, the bill extends funding through FY2022 for state health insurance programs, area agencies on aging, aging and disability resource centers, technical assistance related to outreach and enrollment, and certain quality-measurement activities; provides statutory authority for the Centers for Medicare & Medicaid Services transition program under the Medicare prescription drug benefit for limited-income, newly eligible individuals (i.e., individuals who are eligible for subsidies and have not yet enrolled in, or whose coverage has not yet taken effect under, a Medicare or Medicare Advantage prescription drug plan); revises payment rules for graduate medical education costs with respect to a hospital that establishes a new medical residency training program; and authorizes additional quality measures for the skilled nursing facility (SNF) value-based purchasing program, which provides incentive payments to SNFs based on their quality of care.