I wish I would have been introduced to Geometric Algebra or calculus of forms or whatever it is called during my physics studies. We learned all the conventional things you need for classical mechanics and electromagnetism, like div and curl and BAC-CAB. But there were a couple of things that we were not tought well, which caused problems later. One thing is that at first, a vector was just an N-tuple. But in physics, something is only a vector or a tensor if it behaves under a very specific way during transformations. (The infamous "a tensor is an object that transforms like a tensor"...)

The other thing is that I hit a wall when trying to read theory papers, because nobody ever explained what a wedge operator or a two-form and so on is. I was able to mechanically follow calculations, but never got a real good intuition.

We learned math together with the mathematicians in a very axiomatic way. Without a geometric intuition, I had a hard time understanding why you need dual vectors like one-forms or covariant vectors. They seem just like a convoluted way to write the scalar product.