The F1 visa is a legal requirement for all persons wishing to study at an American higher educational institution. This visa will be valid for the full duration of your studies (you may be asked to renew your visa about 20 months in). An ESTA is a tourist waiver visa allowing you to holiday in the USA for a maximum of 90 days per year, it is therefore not accepted by Immigration and Customs.
You will not need to request an ESTA. The F1 visa will be sufficient.
The visa costs 350 USD for the SEVIS fees and 185 USD for the F1 visa.
