The National Museum of the Air Force tells the story of the U.S. Air Force in its many iterations. But the story of the Air Force is pretty much also the story of flight as both histories overlap for the most part.
If we go back to the late 1400's and early 1500's, there is the story of Leonardo Da Vinci. He was an artist and also a bit of an inventor. He conceptualised some elements of flight.
To get to an actual flight, we have to go to 1783 in France. The brothers Joseph and Jacques Montgolfier were inventors. Well Joseph was the inventor and Jacques was the business end (there is a very funny Monty Python sketch called the Golden Age of Ballooning). They ran a paper mill. In 1782, they made a prototype of the device that would later take them up in the air. So they are credited with the invention of the hot air balloon.