The main point of flight school is obvious; your flying skill increases. To get Gold medals in the challenges, particularly the later ones, you will need a good pilot. Trevor is the obvious choice unless you have built up the others’ flying skill to maximum. Without a maximum flying skill, the latter challenges become very difficult to get Gold in because of added turbulence and harder landings. Helicopter flying is particularly hard.
If you are just aiming to complete Flight School, then choose a character who needs the lessons because it will significantly increase their skill. Get a Bronze or better to unlock the subsequent challenge. There are 12 challenges in all to complete, each one getting more complex as you progress. Speed is the key to getting a Gold medal so be efficient and yet careful during these challenges.