Actually, Newtonian mechanics is "wrong", but not by a long shot. Special Relativity is more correct (we'll ignore GR for now), but the corrections are on the order of (v/c)^4 (using expressions for kinetic energy as a benchmark).

Academia held on to Newton's theory because
1)It is close enough for the vast majority of earth bound applications without the added complexity to get that 14th decimal corrected.
2)It is a simple and intuitive theory that provides a good jumping off point for non-intuitive theories like quantum mechanics and relativity.

AMEN!

I been teaching (about 18 years) freshman/sophomore level physics, primarily engineering students. I try to encourage study groups outside of class and my most successful years as a teacher are when the students are successful in forming these groups.

I also subscribe to the "see it, do it , teach it " philosophy of learning where you develop the deepest understanding of materials when you are forced to explain it to someone else. I use this argument on my better students and the result generally is better performance all around.