I've just started with the book "Modern Classical Physics" by Blandford and Thorne. It has the interesting premise of explaining classical physics from a (coordinate-free where possible) geometric viewpoint. One thing that always confuses me is that there are so many different ways folks present geometric frameworks.
I scoped out whether some terms were present or not. Tensors form an essential part of the book, and yet tensors are defined as "rank $n$" objects that map $n$ vectors to a real number. There's no mention of covectors/dual vectors, multilinear maps or expression of tensor rank as $(m,n)$.
So before I invest my time in the 1500-page book, I'd like some guidance from the community here - how essential is the concept of dual vectors (not just classical, but beyond as well since I'll end up applying these concepts in any further learning I do) in Physics? Is the notion of tensors with just one number instead of a pair of numbers specifying the rank too limited, or is it perfectly okay? Is it important to make the distinction between vectors and dual vectors in Physics fields - will I be severely limiting myself if I can't make that distinction?
It really depends on which kinds of physics you are interested in. For more theoretical and geometrical branches of physics yes, you want the full notion of a tensor. A possible definition of a tensor is as a multilinear map from $k$ copies of a vector space $V$ and $l$ copies of its dual $V^*$ to the field of scalars, usually $\mathbb{R}$. That is where the pair of numbers you refer to is coming from.
If you have an inner product on $V$ you get a canonical isomorphism between $V$ and $V^*$ so you can blur the distinction between the two of them and that is probably what the book is doing.
The concept of a dual vector is pretty important, and it is a special case of a tensor I'd definitely recommend learning about it.