Flying lessons would be starting on Thursday — and Gryffindor and Slytherin would be learning together.
In Harry Potter and the Philosopher's Stone, Chapter Nine - The Midnight Duel Harry's first -
At three-thirty that afternoon, Harry, Ron, and the other Gryffindors hurried down the front steps onto the grounds for their first flying lesson.
(and only) flying lesson is described. During this single lesson Harry discovers that he has a tremendous natural ability to fly on a broomstick and is made Seeker of the Gryffindor Quidditch team.
Flying lessons are never mentioned again in the first book, and as far as I recall, not a single other time in the entire seven books. Harry clearly didn't require them but there were a lot of other students who would have benefited from them, including Hermione.
What amount of tutoring is actually available for those wishing to learn to fly at Hogwarts? Are flying lessons mandatory for first years? Do they continue in subsequent years as well?