Right Start Child Care and Education Act of 2013 - Amends the Internal Revenue Code to: (1) increase the rates and maximum allowable amount of the tax credit for employer-provided child care facilities; (2) increase the eligibility threshold amount and rate of the household and dependent care tax credit and make such credit refundable; (3) allow a new $2,000 tax credit for child care providers who hold a bachelor's degree in early childhood education, child care, or a related degree and who provide at least 1,200 hours of child care services in a taxable year; and (4) increase the tax exclusion for employer-provided dependent care assistance.