The Flight Test side quest in Hogwarts Legacy can be started after you attend flying class for the first time. Travel to Spintwitches in Hogsmeade and purchase a broom from Mr. Weekes (they all cost 600 gold). This will prompt him to ask you if you’d be willing to test out some upgrades to the broom for him so that he can tweak them. Agree to help head over to the Quidditch Pitch on the west side of Hogwarts.
Talk to Imelda Reyes there to start the broom trial. You’ll need to fly through all 22 rings in 2 minutes and 20 seconds to win. There are also yellow bubbles you can fly through along the way which will increase your speed. If you miss any rings, time will be added to your overall time at the end.
After clearing the trial, speak with Imelda again then return to Mr. Weekes to let him know what happened. This will complete the quest.