Rural Educator Support and Training Act or the REST Act This bill amends the Higher Education Act of 1965 and the Elementary and Secondary Education Act of 1965 to: (1) establish several new grant programs for students training to become educators in rural schools, and (2) enhance federal student loan forgiveness for rural educators. The Department of Education (ED) shall award undergraduate scholarships and graduate fellowships to students who are studying to become educators in rural schools. To be eligible, a recipient must maintain acceptable academic standing and shall agree to fulfill a service obligation of three or more years, depending on the number of years for which funding is provided. ED may also award grants to eligible educational agencies for the purpose of: (1) reimbursing eligible rural educators for out-of-pocket costs associated with obtaining national board certification, and (2) increasing annual compensation for eligible rural educators who have become certified. Rural educators shall also be eligible for additional student loan forgiveness. Specifically, a teacher who is employed for five consecutive years in a rural school shall be eligible for up to $17,500 in federal student loan forgiveness. Under current law, a highly qualified teacher who is employed for five consecutive years in a high-poverty school is eligible for up to $5,000 in federal student loan forgiveness; such a teacher who teaches mathematics, science, or special education is eligible for up to $17,500.