This is my askhistorians debut, so please let me know if I should change anything.

The fighter role as we know it today was born during the First World War. The very first fighter pilots were created by the needs of the war. As the war wore on, nations created more rigid flying schools, although specialized instruction would not emerge until the interwar period. By 1914 the modern airplane (the Wright Model A) had been around for only about a decade, and most pilots were civilians who saw flying as a new sport. The Wright brothers believed the airplane held strategic value, and so they approached several nations about purchasing their aircraft, including Britain, Germany, France, and the United States. The United States created the world's first aeronautical division (specifically heavier-than-air) in 1907 and purchased the Model A in 1909. The Wright brothers actually trained some of the military's pilots. In other nations, the military recruited other civilian pilots to join as enlisted pilots or to train their new pilots.

In the first months of the Great War, airplanes were used strictly for reconnaissance. As most planes were two-seaters, the pilot would simply fly the plane while an officer would observe and record artillery emplacements, troop movements, trench structure, and so on. It was more advantageous to use airplanes than balloons for reconnaissance, as the observation balloons could be and often were destroyed by artillery and hostile ground fire. Planes could move quickly around the battlefield and were much harder to shoot down. Obviously it became important to deny these advantages to the enemy. Observers were then given a rifle and/or pistol and told to shoot down other reconnaissance aircraft. This is difficult to do, so some pilots experimented with attaching machine guns to their planes. French aviator Roland Garros succeeded in this respect by adding metal deflectors to his plane's propellers. He went on to shoot down a German recon plane on April 1, 1915. The creation of the interruptor gear, or synchronizer, really established the role of fighter-scout. These pilots' primary goals were to protect friendly reconnaissance aircraft and destroy enemy aircraft. Oswald Boelcke is an example of a pilot who started out flying scouting missions but would go on to become a top ace.

The casualty rates for pilots were very high (roughly 15-25% for both Allied and German pilots), so aerial divisions needed to recruit and train men quickly. Many pilots were former cavalry officers who decided to try their hand at flying after the decline of cavalry divisions. Flight schools were not especially rigid as they would be in World War II; the prevailing theory was that fighter pilots needed to be individualistic, passionate, independent young men. The training was relatively homogenous--only the French identified flight training by special roles (bombing, reconnaissance, etc). Their method was inefficient for the needs of World War I, but would form the basis of flying schools during the interwar period. British schools were composed of a ground section for pilots and observers, where they would learn basic theory and mechanics. If the pilot qualified from there, he would go on to receive individual instruction followed quickly by solo flights. This is where most washouts occurred. By the end of the war, British pilots usually had about 50 solo flight hours before shipping out. Those who would go on to become fighter pilots were trained in advanced maneuvering by their squadron leaders who were usually veterans of the front. Again, Oswald Boelcke was instrumental in establishing the theory of dogfighting and would be the mentor of Manfred von Richthofen (The Red Baron).

During the interwar period and into the first years of the Second World War, flight schools recognized the need for specialized pilots. Potential candidates were selected from the basic rank-and-file or recruited from university. Recruits usually went through basic training where they learned flight theory, navigation, basic military indoctrination, and so on. From there they were split into specializations. Those who were selected to fly fighters were trained in advanced maneuvering, gunnery, and dogfighting. That is essentially how it would stay through World War II.

If I need to add anything or if there are any questions let me know.

Sources
Hampton, Dan. Lords of the Sky: Fighter Pilots and Air Combat, from the Red Baron to the F-16. (New York: HarperCollins, 2014).
Hallion, Richard P. Rise of the Fighter Aircraft: 1914-1918. (Annapolis: 1984).