I'm convinced that it's only a historical accident that we teach people classical logic before intuitionistic logic. CL is terrible.
Everyone is baffled by the CL claim that (A implies B) is equivalent to (A is false or B is true). And they should be, it makes no sense at all. Logic is supposed to be an abstraction of ordinary plausible reasoning, and the equivalence of A→B and ¬A∨B is neither ordinary nor plausible.
Instead IL asks: what do we want A→B to *do*? The answer is simple: If we want to prove B, then knowing A→B should mean that it suffices to prove A instead. That's simply modus ponens, most ordinary and plausible of all logical principles.
And that's exactly what A→B *does* mean in IL, nothing more. How do we prove A→B? By showing that A is false or B is true? No, ridiculous. We do it by demonstrating that we can take a proof of A and convert it into a proof of B—which is precisely what we wanted it for: if we want to prove B it now suffices to prove A instead and then convert the proof.
More generally, CL claims to be concerned with “truth”, whatever that is. Truth is super-complicated, something philosophy has struggled with for thousands of years, and is mixed up with the actual state of the world. But logic shouldn't be concerned with the actual state of the world. It's only trying to formalize ordinary plausible reasoning, not whether snow is white.
IL is much less ambitious. IL judgments are only about provability, which is a much more circumscribed notion than truth. Teaching CL to beginners is like trying to tackle proof theory and model theory at the same time.