Classical mechanics are very important for everyday physics. For the energy scales, relative velocity differences, and mass scales that we experience is our everyday lives, Newtonian physics provide us with an extremely valuable tool of predicting outcomes of events. In other words Newtonian physics are an accurate enough approximation to the more precise theory, special relativity.
Lets not forget that Newtonian physics is accurate enough to take us to the moon!
Other than that some conservation laws from Newtonian physics carry on to the rest of physics, for example conservation of energy and momentum. A young physicist would therefore need a lot of experience with applying these laws, and the best way to do this is with the more 'intuitive' Newtonian physics.