I learned about differential forms in my physics degree, and then later discovered that the exterior algebra made a lot more sense if you introduced it separately. Basically it seems like the exterior algebra is useful in many settings, and differential-forms are just one rather confusing application of it. My current opinion is that exterior algebra should, someday, be taught first (around the time of linear algebra), and then later extended to differential forms when derivatives and manifolds get involved.
I agree that the Hodge Dual is easily the worst part of exterior algebra. But you can treat the inner product as more fundamental, via *a ∧ b = <a,b> i. Either can essentially be constructed from the other (iirc).
On it's own it's not bad, but it's troublesome because given a complicated expression with wedges and stars, it's not at all intuitive how you can simplify it without working in coordinates.
Is there a silver bullet, though? The complexity of having to mess around with epsilon tensors is still there no matter which abstraction you use to sweep it under the rug...
I agree that the Hodge Dual is easily the worst part of exterior algebra. But you can treat the inner product as more fundamental, via *a ∧ b = <a,b> i. Either can essentially be constructed from the other (iirc).