You must submit an online application through the Fulbright portal. Some universities have Fulbright committees and you may first be interviewed through them to be assessed. From there, your application goes through two reviews. One is at the national level for all Fulbright applicants and the second one is by the Fulbright national foundation for the country you applied for. Teaching experience is a plus, but there were many ETAs who had not taught before. Some also just had tutoring experience in a one-on-one situation.