Advancing FASD Research, Prevention, and Services Act - Amends the Public Health Service Act to revise and extend the Fetal Alcohol Spectrum Disorders (FASD) programs (as renamed under this Act, previously the Fetal Alcohol Syndrome and Fetal Alcohol Effect programs). Directs the Secretary of Health and Human Services (HHS) to: (1) establish and carry out a research agenda for FASD; (2) facilitate surveillance, public health research, and prevention of FASD; and (3) continue the Interagency Coordinating Committee on Fetal Alcohol Syndrome. Requires the Secretary to provide financial assistance to: (1) establish or expand state FASD programs; (2) implement best practices to educate children with FASD, educate members of the criminal justice system on FASD, and educate adoption or foster care agency officials about services for children with FASD; (3) provide transitional services for those affected by prenatal alcohol exposure; (4) develop public service announcements to raise awareness of the risks associated with alcohol consumption during pregnancy; (5) increase awareness and identification of FASD in federally qualified health centers; and (6) provide respite care for caretakers, recruit mentors, and provide educational and supportive services to families of individuals with FASD. Directs the Comptroller General to make recommendations regarding the appropriate roles and responsibilities of federal entities with respect to programs and activities related to FASD. Requires the Secretary of Education to direct the Office of Special Education and Rehabilitative Services to conduct training on FASD surveillance and disseminate best practices in the education and support of children with FASD. Requires the Attorney General to direct the Office of Juvenile Justice and Delinquency Prevention to: (1) implement FASD screening procedures; (2) conduct training on surveillance and on identification and support of individuals with FASD; and (3) study the inadequacies of the current system.